Conic Sections- Complete Info

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CONIC SECTIONS TOTAL INFORMATION

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CONIC SECTIONS__CONCEPTS_TO REMEMBER Let S be a fixed point and ‘l ’ be a fixed line. A point P moves such that SP/PM = e., where PM is the r distance from P on to the directrix. Then the locus of P is called a conic. Here, the fixed line L is called directrix and the number e is called eccentricity. If 1) e = 0, then the locus of P is called a circle.

P

M

S (fixed) l (fixed line)

2. e = 1, then the locus of P is called a parabola. 3. 0 < e < 1, then locus of P is called the ellipse. 4. e > 1, then the locus of P is called a Hyperbola. OBSERVATIONS : 1.

A conic is a second degree non-homogeneous equation in x and y.

2.

A second degree non homogeneous equation in x and y, Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0 represents.

1.

Pair of lines : If,  = 0, H2 AB, G2  AC, F2  BC.

2.

Circle : If, 0, A = B, H = 0., G2 + F2 - AC 0

3.

Parabola : If, 0, H2 = AB.

4.

Ellipse : If 0, H2 < AB.

5.

Hyperbola : If 0, H2 > AB.

THE PARABOLA DEFINITIONS, FORMULAE, FACTS ON PARABOLA 1. 2. 3.

If SP/PM = 1, then the locus of P is called a parabola. The standard form of parabola is y2 = 4ax.

Different forms of Parabolas : (i) y2 = 4ax ; a > 0 This is a parabola, whose axes is along x-axes.

Y

X

A

Y 2

(ii) y = 4ax ; a < 0 This is a parabola whose axes is along x-axes. A

>>1 0 This is a parabola whose axes is along y-axes.

X

A

Y

(iv) x2 = 4ay ; a < 0 This is a parabola, whose axes is along y-axis

A

y 2

X

Y

2

(v) (y - ) = 4a(x - ) (or) x = ly + my + n. This is a parabola, whose axes is parallel to x-axes.

X

A

x

A

(vi) (x - )2 = 4a(y - ) (or) y = lx2 + mx + n. This is a parabola,

y

Y

whose axes is parallel to y-axes. A A

x

Parabola is not a closed curve. Axes of the parabola is the line where it is symetrical about it. Y

M

P

L Q

Z

S

A

L

l

- - - - - - - - - - - - - -

4. 5. 6.

X



y

E

X

y

T R

From the above diogram 1.

The line ZM r to x-axes (Axes of the parabola) is called directrix.

2.

The line which is r to the axes and passing through the focus S is called latus rectum. From the diagram, LL is latus rectum.

3.

The chord which passes through the focus of the parabola is called focal chord. From the above diagram, QR is focal chord.

4.

The line passing through P and r to axes of the parabola is called double ordinate. From the above diagram, PT is double ordinate. >>2 0, S11 < 0, S11 = 0 respectively.

9.

Equation of the tangent to the parabola at P(x1, y1) is yy1 - 2a(x + x1) = 0.

10.

Condition at which the line y = mx + c is tangent to the parabola y2 = 4ax is c = a/m

11.

Equation of any tangent to the parabola y2 = 4ax is y = mx + (a/m)

12.

If the line y = mx + c is tangent to the parabola y2 = 4ax, the point of contact is (c/m, 2a/m) (or) (a/m2, 2a/m)

13.

The condition at which the line lx + my + n = 0 may be a tangent to the parabola y2 = 4ax is ln = am2

14.

Point of contact of the line lx + my + n = 0 w.r.t the parabola y2 = 4ax is

15.

Equation of the normal to the parabola y2 = 4ax is y = mx - 2am - am3. Where m is slope of the normal to the parabola.

16.

Atmost two tangents are possible to draw to a parabola.

17.

Equation of pair of tangents to the parabola y2 = 4ax at (x1, y1) is S12 = S.S11

18.

Equation of the chord joining the points (x1, y1) and (x2, y2) on the parabola y2 = 4ax is

FG n , Hl

- 2am l

IJ K

S1 + S2 = S12 (or) (y - y1) (y - y2) = y2 - 4ax. 19.

The locus of point of intersection of r tangents to the parabola is directrix

20.

The line y = mx + c is tangent to the parabola y2 = 4a(x + a) then the condition is c = am + (a/m)

21.

Locus of the foot of the r from the focus to the tangent of the parabola y2 = 4ax is tangent at the vertex (i.e., y-axes).

22.

[The locus of the foot of the r from focus on a tangent to along the parabola]. Equation of the chord of contact of the point (x1, y1) w.r.t. the parabola y2 = 4ax is yy1 - 2a(x + x1) = 0.

23.

Equation of the polar w.r.t the point P(x1, y1) to the parabola y2 = 4ax is y y1 - 2a(x + x1) = 0

24.

Pole of the line lx + my + n = 0 w.r.t the parabola y2 = 4ax is

25.

If the lines l1x + m1y + n1 = 0, l2x + m2y + n2 = 0 are conjugate w.r.t. the parabola y2 = 4ax, is n2 l1 + n1 l2 = 2am1m2. >>3>4>5>6>7>8>90, to that the area of the bounded region enclosed between the parabolas y  x  bx 2 and y 

66.

x2 is maximum. b

[1997]

Points A, B and C lie on the parabola y 2  4ax. The tangents to the parabola at A, B and C taken in pairs, intersect at points P, Q and R. Determine the ratio of the areas of the triangles ABC and PQR. [1996]

>>10 2.

>>11>12>13>14>15>16 2a

2 (h – 2a), 0 3

but

2 (h – 2a) > 0 3

Hence abscissa of the point of concurrency of 3 concurrent normals > 2a. Prob.1 Find the locus of a point which is such that (a) two of the normals drawn from it to the parabola are at right angles, (b) the three normals through it cut the axis in points whose distances from the vertex are in arithmetical progression. [Ans : (a) y2 = a(h – 3a) ; (b) 27ay2 = 2(x – 2a)3 ] Sol.

(a)

we have

m1 m2 = – 1

also

m1 m2 m3 = –

k a



m3 =

put m3 = –

k a

k is a root of a

am3 + (2a – h)m + k = 0

>>17>18>19>20>21 2a, t1 + t2 + t3 = 0 k = 0  x–axis is a normal h = 3a, k = 0  two normals orthogonal.

>>22>23>24>25