Ellipse 2012

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x2 y2 + 2 = 1 , (a < b) 2 a b The graph of this ellipse as shown below.

Believe on your strength

Other Form of the Ellipse :

Chapter 8. ELLIPSE Definition : The locus of a point P whose distance from a fixed point S bears a constant ratio e (e < 1) to its distance from the fixed line d, is called as an ellipse. The fixed point and fixed line respectively are called as the focus and directrix of the ellipse and the constant ratio e is called as the eccentricity of the ellipse.

Y

↑

↑

A|

Z|

→

S

A

B|

O

→ X

A

S| B| →

→ d|

↑ B

d

Z

→ X

x2 y2 x2 y2 + 2 = 1 ,(a>b) 2 + 2 = 1, (a < b) 2 a b a b

Equation

↑

↑

↑

A|

Comparative Study Of Ellipses :

O

S|

d

S

B

→

→ B

Standard Equation of Ellipse : The equation of ellipse in 2 2 standard form is x2 + y 2 = 1 , (a > b). The graph of the a b standard form ellipse is as below | ↑ d

↑

→

Centre

A(a, 0), A (-a, 0)

B(0, b), B|(0, -b)

Foci

S(ae, 0), S|(-ae, 0)

S(0, be), S|(0, -be)

b2 = a2(1 - e2)

a2 = b2(1 - e2)

Relation bet-

as well as the origin. Also origin bisects every chord

ween a, b & e

|

Directrices

a a x= e &x=e

Equation of

major axis: y = 0

major axis: x = 0

minor axis : x = 0

minor axis: y = 0

is called as the centre of the ellipse. Since the ellipse is symmetic about y-axis, it has two foci

(0, 0)

Vertices 2 2 Nature of the Ellipse : x2 + y 2 = 1 , (a > b) a b 1. This ellipse is symmetric about both the coordinate axes

through it, therefore this ellipse is a central conic. The origin

(0, 0)

y=

b b &y=e e

and two directrices. The lines about which the ellipse is

Axes

symmetric are called as the axis of symmetry or simply axes

Length of the

major axis = 2a

major axis = 2b

of the ellipse . The axis of the symmetry for this ellipse are

axes

minor axis = 2b

minor axis = 2a

both x-axis and y-axis. x axis is called as the major axis and

2

Length of L.R.

2b a

2a 2 b

Focal distance

SP = a - ex1

SP = b - ey1

y as the minor axis . 2. This ellipse intersects the coordinate axes at the four |

|

points A(a, 0), A (-a, 0), B(0, b) and B (0, -b). Generally the

|

of P(x1, y1)

S|P = b + ey1

S P = a + ex1

intersection of the ellipse with major axis are taken as the vertices of the ellipse. Here the vertices are A(a, 0) and

Extrimities of (ae,

A|(-a, 0).

Latus-recta

3. The foci (plural of focus) of this ellipse are S(ae, 0) and a S|(-ae, 0). The corresponding directrices are x = and e -a x= respectively. e 4. The curve lies completely in the rectangle bounded by the

ween foci

lines x = + a and y = + b.

Distance betw-

Some Important Definitions :

een dirctrices Parametric equations

(-ae, Distance bet-

1. Focal distance : The distane of a point P on the ellipse from the focus is called as the focal distance of that point. Since their are two foci, therefore there are two focal

b2 b2 ) , (ae, - ) a a

(

a2 a2 , be), (- , be) b b

b2 b2 a2 a2 ) , (-ae, - ) ( , -be), (- , -be) a a b b 2ae 2a e x = acosθ y = bsinθ

2be 2b e x = acosθ y = asinθ

Parametric Equations of the Ellipse : The parametric equations of the ellipse are x = acosθ, y = asinθ , where θ

distances for P given by SP = a - ex1 and S|P = a + ex1,

is called as the parameter and therefore any point on the

where x1 is the abscissa ( x- coordinate) of P.

ellipse can be written as P(θ) = (acosθ, bsinθ)

2. Length of the Latus rectum :- The chord perpendicular to

Director Circle : The locus of point of intersection of perpendicular tangents drawn to the ellipse is called as the director circle. The director circle of the ellipse

the major axis through the focus is called as the latus rectum. The length of the latus rectum for the standard 2b2 ellipse is . a

x2 y2 + = 1 is x2 + y2 = a2 + b2 a2 b2

1

Special Form of the ellipse : 1.The equation of the ellipse

Tangent in terms of Slope : The line y = mx + c touches x2 y2 the ellipse 2 + 2 = 1 , if a b c2 = a2m2 + b2

with centre at (h, k) and the axes parallel to the coordinate (x - h)2 (y - k)2 axes is + =1 2 a b2 2. The equation of the ellipse whose focus is the point (f, g),

and the equation of tangent in terms of slope is

directrix is ax + by + c = 0 and eccentricity is ‘e’, is (ax + by + c)2 (x - f)2 + (y - g)2 = e2 a2 + b2 Auxilliary Circle : The locus of foot of the perpendicular

y = mx + √a2m2 + b2 a2m b2 and the point of contact is (, ) where c = √a2m2 + b2 c c Remarks :

from the focus on any tangent to the ellipse is called as the

1.Since the radical sign on the right side side may have

auxilliary circle or it is also the circle drawn on the major

positive or negative value, therefore there are two tangents

axis as the diameter. The equation of auxilliary circle for the

to the ellipse with same value of m and which are parallel to

2

2

2

standard ellipse is x + y = a .

each other.

General second degree equation : The general second 2

2. The line xcosα + ysinα = p touches the ellipse, if

2

degree equation in x and y, i.e. ax + 2hxy + by + 2gx +

p2 = a2cos2α + b2sin2α

2fy + c = 0 represents an ellipse, if

3. The line lx + my = n touches the ellipse, if

i) abc + 2fgh - af - bg - ch ≠ 0 and ii) h < ab 2

2

2

2

n2 = a2 l2 + b2 m2

Position of a point relative to an Ellipse :

4. If m1 and m2 are the slopes of the tangents drawn to the x2 y2 ellipse 2 + 2 = 1from the point (x1, y1), then m1 and m2 a b are the roots of the quadratic equation

The point (x1, y1) lies within, upon or outside the ellipse x2 y2 depending upon the quantity 1 + 1 - 1 is negative, zero a2 b2 or positive respectively.

m2(x12 - a2) - 2mx1y1 + (y12 - b2) = 0 2x y y 2 - b2 ∴ m1 + m2 = 2 1 1 2 and m1m2 = 12 2 x1 - a x1 - a Tangent in parametric form : The equation of tangent to x2 y2 the ellipse 2 + 2 = 1, at the point whose eccentric angle a b is θ, is xcosθ ysinθ + =1 a b Intersection of two tangents :

Length of the Radius vector : The length of the radius vector r of any point on the ellipse (i.e. distance of any point from the centre) whose inclination is θ is given by a2b2 r2 = b2cos2θ + a2sin2θ Line joining two points on the Ellipse : The equation of the line joining the points P(θ) and Q(φ) is

The coordinates of point of intersection of two tangents xcosθ ysinθ xcosφ ysinφ + = 1 and + = 1 are a b a b

y θ −φ x θ+φ θ+φ ) ) = cos( cos( ) + sin ( b 2 a 2 2 Intersection of Ellipse and line :

x = a.

The x-coordiantes of the point of intersection of the line x2 y2 y = mx + c and the ellipse 2 + 2 = 1 are the roots of the a b quadratic equation

cos 12 (θ + φ) cos 1 (θ - φ)

, y = b.

2

cos 12 (θ + φ) sin 1 (θ - φ) 2

Diameter of an Ellipse : The locus of the middle points of parallel chords of an ellipse is called a diameter, and the

x2(a2m2 + b2) + 2a2mcx + a2(c2 - b2) = 0 The ordinates can be obtained by putting these values of x

chords are called the double ordinates.The diameters pass

in the equation of line.

through the centre O, therefore the equation of diameter is in the form y = mx.

Thus the straight line y = mx + c meets the ellipse in two points real, coincident or imaginary according as

Conjugate diameters : Two diameters are said to be

c2 is < = or > a2m2 + b2.

conjuagate when each bisects all chords parallel to the other.

Length of the intercept : The length of the intercept made

The two diameters y = m1x and y = m2x are conjugate, if 2 m1m2 = - b2. a Properties of Conjugate diameters :

by the ellipse on the line y = mx + c is 2ab √(1 + m2) √a2m2 + b2 - c2 a2m2 + b2

1. The tangents at the extrimity of any diameter is parallel to

Tangent to Ellipse :

the chords which bisects it.

1. Tangent at (x1, y1) : The equation of tangent to the ellipse at (x1, y1) of it is xx yy1 1 + 2 =1 2 a b

2. The tangents at the ends of any chord meet on the diameter which bisects the chord. 3. The eccentrc angles φ and θ of the ends of the cojugate diameters differ by π/2. i.e. φ - θ = + 900.

Important : The equation of tangent to the ellipse (also to any curve) is obtained from that of the equation of the ellipse (or the curve) by substituting xx1 for x2, yy1 for y2, x + x1 for 2x and y + y1 for 2y

Important : The point of intersection of ellipse and 2 the line y = mx + c is b2 ( 2-a m2 2 , ) √b + a m

2

√b2 + a2m2

MULTIPLE CHOICE QUESTIONS

4. The sum of squares of any two semi-conjugate diameters of an ellipse is constant and equal to the sum of the squares

1. In case of the ellipse 9x2 + 25y2 = 225, the lengths of

of semi-axes of the ellipse.

major and minor axes are:

5. The tangents at the ends of a pair of conjugate diameters of an ellipse form a parallelogram whose area is equal to the product of the lengths of the axes.

a) 9,25

b) 25,9

c) 5,3

d) 10,6

2. The equation of the ellipse is 4x2 + 3y2 = 12. The eccentri-

Normal to the Ellipse :

city of the ellipse is

2 2 1.An equation of the normal to the ellipse x2 + y2 = 1 at a b the point (x1, y1) of it is

a2y1(x - x1) = b2x1(y - y1)

a) 1

b) 1/2

c) 2/3

d) 3/4

3. The equation of an ellipse is x2/25 + y2/49 = 1. The coordi-

2. The equation of the normal at the point whose eccentric angle is θ, is ax.secθ - by.cosecθ = a2 - b2

nates of the foci are

3. Equation of normal in the form y = mx + c is 2 2 y = mx - a -b √a2m2 + b2

a) ( + 2√6, 0)

b) ( + 2√12, 0)

c) (0, + 2√6 )

d) (0, + √6 )

4. The eccentricity of an ellipse is 16x2 + 25y2 = 400 is

Chord of Contact : The equation of chord of contact of

a) 3/5

b) 4/5

tangents drawn from the point (x1, y1) is xx1 yy1 + 2 =1 2 a b

c) 1/2

d) 1/7

5. The equation of an ellipse is 3x2 + 4y2 = 12. The length of the latus rectum is

Chord with given midpoint : The equation of chord of

a) 1/2

b) 2/3

ellipse bisected at the point (x1, y1) is T = S1

c) 3

d) 5/8

where

6. The sum of the focal distances of a point on the ellipse

x12 y12 xx1 yy1 + -1 + 1 T = a2 and S1 = a2 b2 b2

x2/4 + y2/3 = 1 is

Combined equation of pair of tangents : The combined equation of the pair of tangents from the point (x1, y1) is x2 y2 SS1 = T2, where S1, T as above and S = 2 + 2 - 1 a b

a) 2

b) 4

c) 1

d) 3

7. The equations of the directrices of the ellipse : 9x2 + 5y2 = 45

Properties of Ellipse :

a) x = + 9

b) y = + 9

1. The sum of the focal distances of any point on the ellipse

c) x = + 9/2

d) y = + 9/2

8. The equation of the ellipse whose semi axes are 4 and 3 is

is a constant equal to the major axis. 2. The product of the lengths of the perpendicular segments 2 2 from the foci on any tangent to a ellipse x2 + y2 = 1, is b2 . a b 3. The locus of the foot of the perpendicular from a focus of 2 2 the ellipse x2 + y2 = 1,is the auxilliary circle. a b 4. The intercept of the tangent between the point of contact

a) 3x2 + 4y2 = 12

b) 9x2 + 16y2 = 144

c) 4x2 + 3y2 = 12

d) 16x2 + 9y2 = 144

9.The equation of the ellipse whose foci are (0, + 2) and eccentricity 3 is

and directrix subtends a right angle at the corresponding

a) 9x2 + 5y2 = 45

b) 5x2 + 9y2 = 45

c) 5x2 + 4y2 = 0

d) 4x2 + 5y2 = 20

10. The equation of the ellipse having eccentricity 3/5 and dis-

directrix.

tance between foci is 6 is

5. The tangents at the extrimities of the latus rectum meet on the corresponding directrix.

a) 16x2 + 25y2 = 400

b) 9x2 + 25y2 = 225

6. Intercept of the tangent between the tangents at the

c) 25x2 + 16y2 = 400

d) 16x2 + 9y2 = 144

11. The equation of the ellipse with eccentricity 1/√5 and dis-

extrimities subtends right angles at the foci.

tance between directrices is 10 is,

7. N is the foot of the ordinate PN of the point P on the 2 2 ellipse x2 + y2 = 1, and A, A| the extrimities of major a b axis, then PN2 b2 i) = 2 ii) SA.SA| = b2 | AN.A N a 8. The tangents and normals at any point on the ellipse bisect the angle between the focal radii of that point.

a) 5x2 + 4y2 = 20

b) 4x2 + 5y2 = 1

c) 4x2 + 5y2 = 20

d) 5x2 + 4y2 = 1

12.The equation of the ellipse passing through (1,4) and (6, 1) is

Imporatnt: To find the tangent to the ellipse from the point outside it, we generally consider tangent in terms of slope through that point.

a) 3x2 + 7y2 = 115

b) 7x2 + 3y2 = 1l5

c) 7x2 + 3y2 = 21

d) 3x2 + 7y2 = 21

13. The equations of the ellipse having eccentricity √3/2 and passing through (-8, 3) is

3

a) 4x2 + y2 = 4

b) x2 + 4y2 = 100

a) 4x2 + 9y2 + 16x - 54y - 61 = 0

c) 4x2 + y2 = 100

d) x2 + 4y2 = 4

b) 4x2 + 9y2 -16x + 54y + 61 = 0 c) 4x2 + 9y2 +16x - 54y + 61 = 0

14. The equations of the directrices of the ellipse:

d) None of these.

25x2 + 16y2 = 400 are a) 3x = + 25

b) 3y = + 25

c) x = + 15

d) y = + 15

25. The centre of the ellipse 8x2 + 6y2 - 16x + 12y + 13 = 0 is at

15. If the semi- axis of an ellipse are 6 and 5, the equation of the ellipse is 2 2 a) x + y = 1 36 25

2 b) x + 6

y2 = 1 5

a) (1, 1)

b) (- 1, 1)

c) (1, -1)

d) None of these.

26. The foci of the ellipse 25 (x + l)2 + 9(y + 2)2 = 225, are at a) (- 1, 2) and (- 1, - 6)

x2 y2 x2 y2 c) d) + = 1 + = 1 25 36 5 6 16. If the focus of an ellipse is (3,0), eccentricity 1/2 and

c) (- 1, - 2) and (- 2, -1)

b) (- 2; 1) and (- 2,6) d) (- 1, - 2) and (- 1, - 6).

27. The sum of the focal distances from any point on the ellipse 9x2 + 16y2 = 144 is

directrix x = 5, the equation of the ellipse is a) 3x2 + 4y2 = 12

a)32

b) 18

b) 3x2 + 4y2 - 14x + 11 = 0

c) 16

d) 8.

2

2

28. If P(x, y), F1(3, 0), F2(- 3, 0) and 16x2 + 25y2 = 400, then

c) 3x + 4y + 14x -11 = 0 d) 3x2 + 4y2 - 14x -12y+ 11 = 0

PF1 + PF2 , where F1, F2 are the foci, equals

17. The major axis of an ellipse is 3 times the minor axis. Then the eccentricity is a) (2√2)/3

b) 2/5

c) √2/4

d) 1/2

a) 8

b) 6

c) 10

d) 12.

29. If the length of the major axis of an ellipse is two times the length of its minor axis, its eccentricity is

18. The length of latus rectum of an ellipse

x2 y2 + = 1 36 49

is

a) 1/3

b) 1/√3

c) 1/√2

d) None of these

30. The length of the latus rectum of an ellipse is 1/3 of the

a) 72/6

b) 72/7

c) 72/8

d) 72/9

major axis. Its eccentricity is b) √(2/3)

a) 2/3 c) (3/4)4

19. The latus rectum of the ellipse 5x2 + 9y2 = 45 is

d) None of these 2

a) 5/3

b) 10/3

c) 2√5/3

d)√5/3.

31. The line y = 2t meets the ellipse 4x2 + 9y2 = 36 in real points, if

20. Equation to the ellipse, whose focus is (6, 7), directrix is x + y + 2 = 0 and e = 1/√3, is 2

b) | t | < 1

c) | t | > 2

d) | t | < 4

x2 y 2 32. If a > 3 and the eccentricity of the ellipse a2 + 4 = 1 , x 2 y2 √3/2 then the eccentricity of the ellipse 2 + = 1, is a 9 a) 7/16 b) √7/4

2

a) 5x + 2xy + 5y - 76x - 88y + 506 = 0 b) 5x2 - 2xy + 5y2 - 76x - 88y + 506 = 0 c) 5x2 - 2xy + 5y2 + 76x + 88y - 506 = 0

c) √3/2

d) None of these. 21. The eccentricity of the curve represented by the equation 2

a) | t | < 2

d) 2/3

33. A circle is a limiting case of an ellipse whose eccentric-

2

x + 2y - 2x + 3y + 2 = 0 is

ity

a) 0

b) 1/2

c) 1/√2

d) √2. 2

a) tends to 0

b) tends to a

c) tends to b

d) None of these x2 y2 34. The equation + = 1 , represents an ellipse, 2-r r-5 if

2

22. The eccentricity of the ellipse 9x + 5y - 30y = 0 is a) 1/3

b) 2/3

c) 3/4

d) None of these.

a) r > 5

23. Equation of the ellipse whose foci are (4, 0) and (- 4,0)

c) r > 2

d) None of these x2 y2 35. The equation + = 1 , represents an ellipse, 1-r r-3 if

and e = 1/3 is a) x2/9 +y2/8 = 16

b) x2/8 + y2/9 = 16

c) x2/9 + y2/8 = 32

d) None of these.

b) 2 < r < 3

24. Equation to the ellipse whose centre is (- 2,3) and whose semi-axes are 3 and 2 and major axis is parallel to the x-

a) r > 1

b) r < 3

c) 1 < r < 3

d) None of these

36. Let F1 and F2 be the points (0, 4) and (0, -4). The locus

axis, is given by

of the point P such that |PF1| + |PF2| = 6 is

4

a) an ellipse

b) The segment F1F2

c) the st line through F1,F2 d) None of these unit length. Then the equation of the ellipse is b) x2 + 2y2 = 2

c) 4x2 + 20y2 = 5

d) 20x2 + 4y2 = 5 b) no locus if k > 0

c) an ellipse if k < 0

d) None of these

area of the triangle APA| is a) 1 ab 2 c) ab

c) 3/2

d) None of these

respectively, then

40. The eccentricity of the ellipse 3x2 + 4y2 = 24 is a) 1/4

b) 7/4

c) 1/2

d) None of these

a) Q lies inside C but outside E b) Q lies outside both C and E c) P lies inside both C and E

41. The eccentricity of the ellipse 4x2 + 9y2 + 8x + 36y + 4

d) P lies inside C but outside E

= 0 is 3 a) 5 b) 5 6 c) √2 d) √5 3 3 42. The eccentricity of the ellipse whose latus rectum is half

53. The eccentricity of the ellipse whose pair of conjugate diameters are y = x and 3y = -2x is

its minor axis is a) √2/3

b) 1/√2

c) √3/2

d) None of these

a) 1/√3

b) 1/3

c) 2/3

d) None of these

54. An ellipse has OB as a semi-mimnor axis. S1, S2 are its foci and the angle S1BS2 is a right angle, then the eccentricity of the ellipse is

43. The centre of the ellipse 4x2 + 9y2 + 16x - 18y - 11 = 0

a) 1/√2

is

c) 2/3

a) (2, -1)

b) (-2, 1)

d) None of these (x + 1)2 (y + 2)2 44. The foci of the ellipse are + 9 25 = 1 a) (-2, 1) & (-2, 6) b) (-1, 2) & (-1, -6)

a) x2 + y2 - 2y - 8 = 0

d) a - b

+ 2x - 12y + 15 = 0 is a) x2 + y2 + 2x - 6y - 4 = 0 b) x2 + y2 + 2x - 6y + 4 = 0 b) x2 + y2 +2x - 12y + 4 = 0 d) x2 + y2 + 2x - 12y - 4 = 0 57. A man ruuning round a race course notes that the sum of the distances of two flag posts from him is always 10 m. and the distance between the flag posts is 8 meters. The

vertices ( + 5, 0) is x2 y2 x2 y 2 + =1 a) 25 + 9 = 1 b) 25 16 2 2 c) x + y = 1 d) None of these 25 16 47. The curve represented by :

area of the path he encloses in square meters is a) 8π

b) 12π

c) 18π

d) 15π

58. The number of values of c such that the line y = 4x + c touches the curve x2 + 4y2 = 4 is

x = 3(cost + sint), y = 4(cost - sint) is

a) 0

a) a circle

c) 2 `

b) a parabola

c) an ellipse

d) a hyperbola x2 y2 48. Ley P be a variable point on the ellipse 2 + 2 = 1, a b with foci S1, and S2. If A be the area of the triangle

b) 1 d) infinite

59. Chords of an ellipse are drawn through the positive end of the minor axis, then their middle point lies on

PS1S2, then the maximum value of A is

a) a circle

b) a parabola

c) an ellipse

d) a hyperbola

60. The locus of the centre of the circle x2 + y2 + 4xcosθ -

b) abe d) None of these

2ysinθ - 10 = 0 is

49. The equation of the director circle to the ellipse 2

d) None of these

56. The equation of the director circle of the ellipse x2 + 2y2

46. The equation of the ellipse with foci ( + 3, 0) and

a) ab c) e

b) x2 + y2 - 8y + 7 = 0

c) x2 + y2 - 4x - 5 = 0

d) (-1, -2) & (-1, -6) 2 2 45. Sum of the focal distances of an ellipse x + y = 1, 2 2 a b (a > b) is c) a + b

d) None of these

4y2 - 8y - 32 = 0 is

c) (-1, -2) & (-2, -1)

b) 2b

b) 1/√3

55. The equation of the auxilliary circle of the ellipse 9x2 +

c) (-2, -1)

a) 2a

b) 2ab

d) None of these 2 2 x y 52. Let E be the ellipse + = 1, and C be the circle 9 4 2 2 x + y = 9. Let P and Q be the points (1, 2) and (2, 1)

from one end of the minor axis is b) 4

d) x2 + y2 = 125

d) None of these 2 2 51. P is the variable point on the ellipse x2 + y2 = 1, with a b AA| as the major axis, then the maximum value of the

39. The distance of a focus of an ellipse 9x2 + 16y2 = 144 a) 3,

c) x2 + y2 = 325

c) 4

38. The equation 2x2 + 3y2 - 8x - 18y + 35 = k represents : a) a point if k = 0

b) x2 + y2 = 25

50. The radius of the circle passing through the foci of the 2 2 ellipse x + y = 1 and having its centre at (0, 3) is 16 9 a) 3 b) 5

37. The foci of an ellipse are (0, +1) and minor axis is of a) 2x2 + y2 = 2

a) x2 + y2 = 13

2

2x + 3 y = 30 is,

5

a) a circle

b) a parabola

c) an ellipse

d) a hyperbola

x2 y2 x2 y2 a) 16 + 25 = 1 b) 9 + 16 = 1 2 2 c) x + y = 1 d) None of these 25 16 74. The equation of the ellipse pasing through the intersec

61. The equation of standard ellipse whose axes are along the coordinate axes and eccentricity is 1 , is √2 a) x2 + 2y2 = 32 b) 2x2 + y2 = 64 c) x2 + 4y2 = 256

d) 4x2 + y2 = 64

tion of the lines 7x + 13y - 87 = 0 and 5x - 8y + 7 = 0

62. The equation of conjugate diameter of the diameter x2 y2 y = x of the ellipse 2 + 2 = 1, is a b a) a2x + b2y = 0 b) ax + by = 0

and having the length of the latus rectum 23√2/5, is x2 y2 x2 y2 + =1 a) b) + =1 50 32 27 8 2 2 c) x + y = 1 d) None of these 18 12 75. The equation of the ellipse having focus at (3, -3), the

c) b2x + a2y = 0

d) None of these x2 y2 + = 1, is 63. The eccentricity of the ellipse 25 144 a) √117/12 b) √118/12 c) √119/12

corresponding vertex (4, -3) and the centre at (2, -3) is 2 2 (y + 3)2 a) (x + 2)2 b) (x - 1) + (y + 3) = 1 4 3 + =1 4 2 2 2 2 (x 1) (y + 3)2 c) (x + 1) + (y + 2) = 1 d) + =1 9 6 9 25

d) √120/12

64. The difference between the length of the major axis and minor axis of the ellipse 9x2 + 5y2 = 45, is a) 6 + 2√5

b) 6 - 2 √5

c) 6 + 5√2

d) 6 - 5√2

76.The ecentricity of ellipse is 1/2, It’s centre is at origin and one directrix has the equation x = 16 calculate the

65. The equation of the ellipse whose vertices are ( + 5, 0)

distance of P on the ellipse from the associated focus

and length of the latus rectum is 32/5, is a) 25x2 + 16y2 = 400

b) 16x2 + 25y2 = 400

c) 16x2 + 25y2 = 800

d) None of these

with the given directrix if x co-ordinate of P is - 4 is

66. The equation of the ellipse whose distance between

c) 12

d) 13

4x2 + 9y2 - 16x - 54y + 61 = 0 is a) outside the ellipse

b) on the ellipse

c) on the major axis

d) on the minor axis x2 y2 78.The equation of tangent to the ellipse + = 1 5 6 at P (3, -4) is

ellipse 16x2 + 25y2 = 400 is c) on the ellipse

b) 11

77. The position of the point (1, 3) with respect to the ellipse

directrices is 10 and eccentricity is 1/√5 x2 y2 x2 y2 + =1 + =1 a) b) 100 125 100 225 2 2 c) x + y = 1 d) None of these 125 100 67. The position of the point (-1, 2) with respect to the a) outside the ellipse

a) 10

b) inside the ellipse d) None of these

a) 3x - 4y = 1

b) 27x + 16y = 36

c) 27x - 16y = 36

d) 16x + 27y = 36

79.The equation of the tangent to the ellipse x2 + 4y2 = 17

68. If the eccentricity of an ellipse is 5/8 and the distance

at point (1,2) is

between its foci is 10, then its latus rectum is

a) x + 8y = 17

b) 4x + 3y = 17

a) 39/4

b) 10

c) 8x + y = 17

d) x - 8y = 17

c) 35/4

d) 37/4

80. The equation of the ellipse is 4x2 + 9y2 = 72. The

69. If the foci and vertices of an ellipse are ( + 1, 0) and

equation of the line that touches it at (3,2) is

( + 2, 0), then the minor axis of the ellipse is

a) 2x - 3y = 12

b) 3x + 2y = 24

a) 2√5

b) 2√3

c) 3x - 2y = 6

d) 2x + 3y = 12

c) 3√2

d) 5√2

81. If the line x + y + k = 0 touches the ellipse x2 y2 + = 1 , then the value of k is 20 5 a) + 1 b) + 3

70. The equations of the directrices of the ellipse 16x2 + 25y2 = 400 are a) x = + 10

c) + 5

d) + 7 x2 y2 82. The equation of the ellipse is + .. = 1 .The 25 16 equations of the tangents making an angle of 60° with the

b) x = + 6

c) 3x = + 10

d) None of these 2 2 71. The foci and eccentricity of the ellipse x + y = 1, 36 9 a) ( + 3√3, 0); √2/6 b) ( + 2√3, 0); √3/6 c) ( + 3√3, 0); √3/6

major axis

d) None of these

72. The length of the latus rectum and eccentricity of the x2 y2 + = 1, is ellipse 64 39 38 6 a) 39 ; 5 b) 5 ; 7 4 8 38 6 39 5 c) d) 5 7 4 8 73. The equation of the ellipse having focus at (1, -2),

a) y = √3 x + √91

b) y + √3 x = + √91

c) √3 x = y + √13

d) y = √3 x + 91

83. The equation of the tangent to the ellipse 4x2 + 9y2 = 36 which is also perpendicular to the line 3x + 4y = 17 is, a) y = 4x + 6√5

b) 3y + 4x = + 6√5

c) 3y = 4x + 6√5

d) 3y = x + 2√5

“Strength does not come from winning. Your struggles develop your strengths. When you go through hardships and decide not to surrender, that is strength.”

directrix 3x - 2y + 1 = 0 and eccentricity 1/√2 is

6

84. If (3, 4) lies on the standard ellipse, then which of the following point also lie on the same ellipse a) (-3, 4) b) (-3, -4) c) (3, -4) d) All of the above. 85. The equation of the ellipse passing through the point

96.Length of the chord intercepted by the ellipse 4x2 + 9y2 = 1 on the line 9y = 1 is

(-2√5, 2) and having its length of the semi-minor axis 3 is

b) 2√2/3

c) √2

d) None of these

97. The line 2x +3y = 12 touches the ellipse 25x2 +16y2 = 400

x2 y2 x2 y2 b) + =1 + =1 36 9 28 14 2 2 2 2 c) x + y = 1 d) x + y = 1 36 9 28 14 86. The equations of tangents to the ellipse from the point

at the point

a)

a) (3/2, 3),

b) (3, 2)

c) (-3/2, 5),

d) None of these 16 98.If the tangent at the point ( 4cosφ, sinφ ) to the ellipse √11

16x2 + 11y2 = 256 is also a tangent to the circle

(2, -2) are a) y = 2, 4x - 5y = 10

a) 2√2

b) y + 2 = 0, 8x - 5y = 26

x2 + y2 - 2x = 15, then the value of φ is

c) y = -4, 8x - 5y + 26 = 0 d) y = 3 , 6x + 5y = 26 87. The equation of an ellipse having minor axis 4 and the line 3x + 10y = 25 is tangent is, 2 2 2 2 a) x + y = 1 b) x + y = 1 16 4 4 25 2 2 2 x y x y2 c) 25 + 4 = 1 d) 10 + 4 = 1

a) + π/3

b) + π/4

c) + π/2

d) None of these

99. The locus of the midpoint of the portion of the tangents to the ellipse b2x2 + a2y2 = a2b2 between the coordinate axes is

88. An ellipse is described by using an endless string which is

a) b2x2 + a2y2 = 4a2b2

b) b2x2 + a2y2 = 4

c) a2x2 + b2y2 = 4

d) b2x2 + a2y2 = 4x2y2

passed over two pins. If the axes are 6 cm and 4 cm, the

100. If the focal distance of an end of the minor axis of the

necessary length of the string and the distance between the

ellipse reffered to the principle axis is k and the distance

pins respectively in cms are .

between the foci is 2h, then its equation is

a) 6, 2√5

b) 6, √5

c) 4, 2√5

d) None of these.

89.If the straight line y = 4x + c is a tangent to the ellipse x2/8 + y2/4 = 1, then c will be equal to a) + 4

b) + 6

c) + 1

d) + √(132).

b)

line (x/7) + (y/2) = 1 on the axis of x and straight line (x/3) - (y/5) = 1 on axis of y and whose axes lie along the

90. The straight line x + y = a will be tangent to the ellipse 2

x2 y2 =1 2 + 2 k k + h2

x2 y2 =1 2 + 2 k h - k2 2 2 x2 y2 c) x2 + 2y 2 = 1 d) 2 + 2 = 1 k h k k -h 101. The eccentricity of the ellipse, which meets the straight a)

2

x /9 + y /16 = 1, if a =

coordinate axes is :

a) 8

b) + 5

a) √2/7

b) 2√6/7

c) + 10

d) + 6.

c) 3√2/7

d) None of these

91. The number of values of c such that the straight line 2

102. The focal chord of the parabola y2 = 16x is a tangent

2

y = 4x + c touches the curve x /4 + y = 1 is

to the ellipse (x - 6)2 + y2 = 2 then the possible values of

a) 0

b) 1

the slope of this chord are :

c) 2

d) 3

a) 1, -1 2

92.Equation of tangent to ellipse 2x + y = 4 at the point

c) -2, 1/2

d) 1/2, 2 (x + y - 2)2 (x - y)2 103. Centre of the ellipse , is + 9 16 = 1 a) (0, 0) b) (1, 1)

(2, 1) is a) 4x + y = 4

b) x + 4y = 4

c) 4x + y = 1

d) None of these.

93.Equation of the tangent to the hyperbola 2x2 - 3y2 = 6 which b) y = 3x - 5

c) y = 3x + 5 and y = 3x - 5 d) None of these. 2

d) (0, 1)

x2 y2 + = 1 at the 9 5 ends of the latus rectum. Then the area of thequadrilatral

so formed is 2

94.The equation of the tangents to the ellipse 9x + 16y = 144 from the point (2, 3) are a) y - 3 = 0, x + y = 5

c) (1, 0)

104. Tangents are drawn to the ellipse

is parallel to the line y = 3x + 4 is a) y = 3x + 5

b) -1/2, 2

2

b) x - 3 = 0, x - y = 5

a) 27

b) 27/2

c) 27/4

d) 27/55

105. If tangents are drawn to the ellipse x2 + 2y2 = 2, then

c) x + y - 3 = 0, x - y +5 = 0 d) None of these.

the locus of the midpoint of the intercept made by the

95. The angle between the pair of tangents drawn to the ellipse tangents between the coordinates axes is :

3x2 + 2y2 = 5 from tbe pomt (1, 2) is a) tan-1 (12/5)

b) tan-1(6/√5)

c) tan-1 (12/√5)

d) tan-1(12/5).

1 1 + =1 2x2 4y2 x2 y2 + =1 c) 2 4

a)

7

1 1 + =1 4x2 2y2 x 2 y2 d) 4 + 2 = 1 b)

106. The angle between tangents drawn from the point 2

2

(1, 2) to the ellipse 3x + 2y = 5 is 8√5 5√5 a) tan-1( ) b) tan-1( ) 3 3 12√5 15√5 c) tan-1( ) d) tan-1( ) 4 3 107. The line xcosα + ysinα = p is a tangent to the ellipse x 2 y2 + = 1, if a2 b2 a) a2cos2α + b2sin2α = p2 b) a2cos2α - b2sin2α = p2 c) a2sin2α - b2cos2α = p2

d) a2sin2α + b2cos2α = p2

108. P and Q are the feet of the perpendicular from the foci 2 2 S1, S2 of an x + y = 1 on the tangent at any point on 5 3 the ellipse, then product of S1P and S2Q is

a) 3

b) 15

c) 5

d) 10

109. The value of k for which the line x - y + k = 0 touches the ellipse 2x2 + 3y2 = 1 is +√5 +√3 a) 6 b) 5 +√5 +√2 c) d) 7 3 110. The equation of the ellipse with foci at ( + 3, 0) and

a) 27

b) 27/2

c) 27/4

d) 27/55

118. If tangents are drawn to the ellipse x2 + 2y2 = 2, then the locus of the midpoint of the intercept made by the tangents between the coordinates axes is : 1 1 1 1 =1 =1 a) 2 + b) 2 + 4x 2y2 2x 4y2 2 2 2 2 c) x + y = 1 d) x + y = 1 4 2 2 4 119. The locus of the midpoint of the portion of the tangent x 2 y2 to the ellipse 2 + 2 = 1 intercepted between the a b coordiante axes is a2 b2 a2 b2 b) + = 1 + =2 x2 y2 x2 y2 2 2 a b c) 2 + 2 = 4 d) None of these x y x2 y 2 120. If a tangent to the ellipse 2 + 2 = 1 cuts off inter a b cepts of length h and k on the coordinate axes, then the a2 b2 value of 2 + 2 = h k a) 4 b) 1 a)

c) -3

d) None of these

121.If P and Q are two points on the ellipse and their ecentric angles differ by π/2, then the equation of the locus of points of intersection of the tangents drawn from

which touches the line x - y - 5 = = is

P & Q is x2 y2 a) 2 + 2 = 2 a b 2 2 c) x2 + y2 = 1 a b

x2 y2 x2 y2 a) b) + = 1 + = 1 16 9 9 25 2 2 x y c) d) None of these + = 1 25 16 111. The slopes of the tangents drawn to the ellipse 4x2 +

x2 y2 + = 3 a2 b2 2 2 d) x2 + y2 = 4 a b b)

122. The equation of the tangents to the ellipse 4x2 + 9y2 = 72

7y2 = 28 from the point (3, -2) a) 1, 2

b) -1, 3

which are perpendicular to the line 3x - 2y = 5 is

c) 0, 5

d) 0, -6

a) 2x + 4y +12 = 0

b) 3x + 4y + 12 = 0

c) 3x + 5y + 12 = 0

d) 2x + 3y +12 = 0

112. The equation of the tangent to the ellipse x2 y2 + = 1 parallel to the line x + y + 3 = 0 are 144 25 a) x + y = + 11 b) x + y = + 13 c) x + y = + 10

123.A tangent havng slope 2 to the ellipse 3x2 +4y2 = 12, intersects X and Y axes in the point A and B respectively.If O is the origin Then the area of the triangle AOB is

d) x + y = + 26

113.The equation of the tangent to the ellipse x2 + 16y2 = 16 making an angle of 600 with major axis is

a) 3 Sq. Units

b) 4 Sq. Units

c) 5 Sq. Units

d) 6 Sq. Units

124. The equation of the tangents to the 9x2 + 16y2 = 144

a) √3 x - y + 6 = 0

b) √3 x - y - 8 = 0

making equal intercepts on co-ordinates axes is

c) √3 x - y + 7 = 0

d) None of these

a) x + y = + 6

114. The product of the length of the perpendiculars drawn x2 y 2 from foci on any tangent to the Ellipse 2 + 2 = 1 is a b a) a2 b) b2 c) a2 b2

c) x + y = + 3

d) x + y = + 5 x2 y2 125. A tangent having slope -4/3 to ellipse + =1 18 32 intersects the X-axis and the Y axis in the points A and B respectively. If O is the origin then the area of ∆OAB is

d) a2/b2

115.The auxilliary equation of the circle of the standard Ellipse is 2

2

a) x + y = a

2

2

2

2

2

b) x + y = a + b

c) x2 + y2 = b2

d) x2 + y2 = a2 - b2 x 2 y2 116. The director circle of the Ellipse 2 + 2 = 1, (a > b) is a b a) x2 + y2 = a2 b) x2 + y2 = a2 + b2 2

2

2

c) x + y = b

2

b) x + y = + 4

2

2

2

d) x + y = a - b x 2 y2 117. Tangents are drawn to the ellipse + = 1 at the 9 5 ends of the latus rectum. Then the area of the quadrilat-

a) 23 Sq.Units

b) 22 Sq.Units

c) 24 Sq.Units

d) 21 Sq.Units

126. The locus of point of intersection of perpendicular tanx2 y2 gents to the ellipse a2 + b2 = 1 is a) x2 + y2 = a2 b) x2 + y2 = a2 + b2 c) x2 + y2 = b2

d) x2 + y2 = a2 - b2 x2 y2 127. The ellipse 2 + 2 = 1 is a curve symmetric about a b a) x axis b) origin c) y-axis

eral so formed is

8

d) all above

128. On ellipse 4x2 + 9y2 = 1 , the points at which tangents

140. The equations x = a (

parallel to the line 8x = 9y are

represent

a) (1/5, 2/5)

b) ( + 2/5, + 1/5)

c) ( +3/5, + 2/5)

d) None of these

a) a parabola 2

following is true b) 2 d) infinite

a) 00

b) 900

c) 450

d) 600

axis. If STB is an equilateral triangle, then the eccentric-

c) y = 0, y = 6

d) None of these

a) 2a2

b) a2

c) b2

d) 2b2

at (-1, 1) is:

134. Two perpendicular tangents drawn to the ellipse 16x2 + a) x2 + y2 = 9

b) x2 + y2 = 41

c) x2 - y2 = 41

d) x = a/e

[CET-2004]

a) -x + 4y = 25

b) -x + 4y = 5

c) x + 4y = 25

d) x - 4y = 5

146. The sum of the squares of the perpendiculars on any x2 y2 tangent to the ellipse 2 + 2 = 1from two points on the a b minor axis each at distance ae from the centre is

25y2 = 400 intersect on the curve

x2 y 2 + =1 a2 b 2

a) 2a2

from a focus is

b) a2

c) b2

a) a(e + cosθ)

b) a(e - cosθ)

c) a(1 + cosθ)

d) a(1 + 2ecosθ)

d) 2b2 x 2 y2 147. If any tangent to the ellipse 2 + 2 = 1intercept equal a b lengths l on the axes, then l is equal to

136. If C is the centre of the ellipse 9x2 + 16y2 = 144 and S is one focus. The ratio of CS to semi-major axis is a) √7 : 16

b) √7 : 4

c) √5 : √7

d) None of these

a) a2 + b2

b) a2 - b2

c) √a2 + b2

d) √a2 - b2

148. The difference between length of the major axis and the latus rectum of the ellipse is

137. Tangents at the extrimities of latus rec:tum of an ellipse intersect on a) Major axis

b) x = + √5

145) The equation of the tangent to the ellipse x2 + 4y2 = 5

ity of the ellipse is

135. The distance of a point θ on the ellipse

a) y = + 3

dinate axes, then its equation is x2 y2 x2 4y2 + = 1 + = 1 a) b) 16 25 80 5 x2 y2 x2 y2 c) d) + = 1 + = 1 5 20 20 5 144. The sum of the squares of the perpendiculars on any x 2 y2 tangent to the ellipse 2 + 2 = 1from two points on the a b minor axis each at distance √a2 - b2 from the centre is

133. S and T are the foci of an ellipse and B is an end of minor

d) 2/3

d) None of these

the line x + 4y - 10 = 0. If its axes coincide with the coor-

d) c > b

c) 1/2

c) foci are (3, 1) & (-1, 1)

143. An ellipse passes through the point (4, -1) and touches

b) a2m2 > c2 - b2

b) 1/3

b) eccentricity is 1/3

major axis of the ellipse 9x2 + 5y2 - 30y = 0 is

2 2 132. The ellipse x + y = 1and the straight line y = mx + c 2 2 a b intersect in real points only if

a) 1/4

a) centre is (2, -1)

142. The equations of the the tangents drawn at the ends of the

130. The locus of the midpoint of focal chord of an ellipse x2 y2 + = 1, is a2 b2 2 x2 y2 ex x y2 ex a) 2 + 2 = b) 2 - 2 = a a a b a b 2 2 2 2 c) x + y = a + b d) None of these x y x2 y2 131. If + = √2 touches the ellipse 2 + 2 = 1 , then a b a b its eccentric angle of the point of contact is

c) a2m2 > c2 - b2

2

141. For the ellipse 3x + 4y - 6x + 8y - 5 = 0 which of the

from the point (3, 5) is

a) a2m2 < c2 - b2

d) a hyperbola 2

129. Number of real tangents drawn to ellipse 3x + 5y = 32

c) 0

b) an Ellipse

c) a circle 2

a) 1

2bt 1 - t2 ) and y = ;t∈R 1 + t2 1 + t2

b) Minor axis

a) ae2

b) 2ae2

c) 2ae

d) ae

149. The equation of the ellipse whose eccentricity is 1/2

c) Corresponding directrix d) none of these.

and the vertices are (4, 0) and (10, 0) is

138. Product of the perpendicular from the foci upon any x 2 y2 tangent to the ellipse 2 + 2 = 1 is constant and = a b a) 2a b) a2

a) 3x2 + 4y2 - 42x + 120 = 0 b) 3x2 + 4y2 + 40x + 130 = 0 c) 3x2 + 4y2 - 38x - 140 = 0

c) b2

d) none of these. x 2 y2 139. Tangents are drawn to the ellipse + = 1 at the 9 5 ends of the latus rectum. The area of the quadrilateral so

d) 3x2 + 4y2 - 36x - 150 = 0 150. The locus of points of intersection of the tangents at the extrimities of the chords of ellipse x2 + 2y2 = 6 which

formed is

touch the ellipse x2 + 4y2 = 6 is

a) 27

b) 13

a) x2 + y2 = 4

b) x2 + y2 = 9

c) 15

d) 45

c) x2 + y2 = 16

d) None of these

9