Analytic Geometry Formulas

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CARTESIAN/RECTANGULAR COORDINATE SYSTEM

STANDARD EQUATION OF LINES 1.

Point-Slope form Given a point 𝑃1 𝑥1 , 𝑦1 and slope 𝑚.

𝒚 − 𝒚 𝟏 = 𝒎 𝒙 − 𝒙𝟏 2.

Slope-Intercept form Given a slope 𝑚 and 𝑦-intercept 𝑏:

𝒚 = 𝒎𝒙 + 𝒃 3.

Intercept form Given x-intercept 𝑎 and y-intercept 𝑏:

DISTANCE BETWEEN TWO POINTS The distance between two point 𝑃 𝑥1 , 𝑦1 and 𝑄 𝑥2 , 𝑦2 is:

𝒅=

𝒙 𝟐 − 𝒙𝟏

𝟐

+ 𝒚𝟐 − 𝒚𝟏

𝟐

Two-point form Given two points 𝑃1 𝑥1 , 𝑦1 and 𝑃2 𝑥2 , 𝑦2 :

𝒚 − 𝒚𝟏 𝒚𝟐 − 𝒚𝟏 = 𝒙 − 𝒙𝟏 𝒙𝟐 − 𝒙𝟏

EQUATION OF A LINE

SLOPE OF THE LINE The slope of the line passing through points 𝑃 𝑥1 , 𝑦1 and 𝑄 𝑥2 , 𝑦2 is:

𝒔𝒍𝒐𝒑𝒆, 𝒎 =

4.

𝒙 𝒚 + =𝟏 𝒂 𝒃

The angle between lines 𝐿1 and 𝐿2 is the angle 𝜃 that 𝐿1 must be rotated in a counter clockwise direction to make it coincide with 𝐿2

𝒓𝒊𝒔𝒆 𝒚𝟐 − 𝒚𝟏 = 𝒓𝒖𝒏 𝒙𝟐 − 𝒙𝟏

Where: m is positive if the line is inclined upwards to the right. m is negative if the line is inclined downwards to the right. m is is zero for horizontal lines

EQUATION OF A LINE GENERAL EQUATION OF A LINE The general equation of a straight line is:

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0

𝐭𝐚𝐧 𝜽 =

𝒎 𝟐 − 𝒎𝟏 𝟏 + 𝒎𝟏 𝒎𝟐

Lines are parallel if 𝑚1 = 𝑚2 Lines are perpendicular if 𝑚2 =

−1 𝑚1

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DISTANCE FROM A POINT TO A LINE The distance (nearest) from a point 𝑃1 𝑥1 , 𝑦1 to a line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 is:

MIDPOINT OF A LINE SEGMENT The midpoint 𝑃𝑚 𝑥𝑚 , 𝑦𝑚 of a line segment through from 𝑃1 𝑥1 , 𝑦1 to 𝑃2 𝑥2 , 𝑦2 is:

𝒙𝒎 =

𝒙𝟏 + 𝒙𝟐 𝟐

𝒚𝒎 =

𝒚𝟏 + 𝒚𝟐 𝟐

CONIC SECTIONS

𝒅=

Conic sections a locus (or path)that moves such the ratio of its distance from a fixed point (called the focus)and a fixed line (called the directrix) is constant. This constant ratio is called the eccentricity, e of the conic.

𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪 ± 𝑨𝟐 + 𝑩 𝟐

DISTANCE BETWEEN TWO PARALLEL LINES The distance between two parallel lines 𝐿1 ∶ 𝐴𝑥 + 𝐵𝑦 + 𝐶1 and 𝐿2 ∶ 𝐴𝑥 + 𝐵𝑦 + 𝐶2 is:

𝒅=

The term conic section is based on the fact that these are the sections formed if a plane is made to pass through a cone. If the cutting plane is parallel to the base of a cone, the section formed is a circle. If it is parallel to the element (or generator) the section formed is a parabola. If it is perpendicular to the base of the cone, the section formed is a hyperbola. If it is oblique to the base or element of the cone, the section formed is an ellipse.

𝑪𝟐 − 𝑪𝟏 𝑨𝟐 + 𝑩𝟐

DIVISION OF LINE SEGMENT

GENERAL EQUATION OF CONICS 𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 If 𝐵 ≠ 0, the axis of the conic is oblique with the coordinate axes ( i.e. not parallel to X or Y axes). Thus if the axis is parallel to either X or Y-axes, the equation becomes

𝒙𝒑 =

𝒙𝟏 𝒓𝟐 +𝒙𝟐 𝒓𝟏 𝒓𝟏 +𝒓𝟐

𝒚𝒑 =

𝒚𝟏 𝒓𝟐 +𝒚𝟐 𝒓𝟏 𝒓𝟏 +𝒓𝟐

𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

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GENERAL EQUATION OF CONICS

PARABOLA PARABOLA – is the locus of a point that moves such that its distance from a fixed point called the focus is always equal to its distance from a fixed line called the directrix.

𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 From the foregoing equations: If 𝐵2 < 4𝐴𝐶, the conic is an ellipse If 𝐵2 = 4𝐴𝐶, the conic is a parabola If 𝐵2 > 4𝐴𝐶, the conic is a hyperbola Also, a conic is a circle if A=C, an ellipse if A≠C but have the same sign, a parabola if either A=0 or C=0, and a hyperbola if A and C have different signs.

CIRCLE CIRLE – is the locus of a point that moves such that it is always equidistant from a fixed point called the center. The constant distance is called the radius of the circle.

General equation of Parabola (A or C is zero)

r = radius (h,k) = center

C=0

General equation of a Circle (A=C)

𝑨𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 0 or 𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

𝑨𝒙𝟐 + 𝑨𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 or

a = distance from the vertex to focus LR = length of latus rectum

𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

A=0

To solve a circle, either one of the following two conditions must be known: 4. Three point along the circle, Solution: Use the general form 5. Center (h,k) and the radius, Solution: Use the standard form Standard Equation of a Circle 𝟐

Center at (h,k)

𝒙−𝒉

+ 𝒚−𝒌

Center at (0,0)

𝒙𝟐 + 𝒚 𝟐 = 𝒓 𝟐

𝟐

= 𝒓𝟐

For the circle

𝑨𝒙𝟐 + 𝑨𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

−𝑫 −𝑬 𝒉= ;𝒌 = ;𝒓 = 𝟐𝑨 𝟐𝑨

𝑫𝟐 + 𝑬𝟐 − 𝟒𝑨𝑭 𝟒𝑨𝟐

𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 or 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 To solve a parabola, either one of the following two conditions must be known: 1. Three point along the parabola and an axis (either vertical or horizontal), Solution: Use the general form 2. Vertex (h,k), distance from the vertex to focus a and axis, Solution: Use the standard form 3. Vertex (h,k), and the location of the focus. Solution: use the standard form Eccentricity The eccentricity of a conic is the ratio of its distance from the focus d2 and the directrix d1 For a parabola, the eccentricity is equal to 1.

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Latus Rectum Latus rectum is the chord passing through the focus and parallel to directrix or perpendicular to the axis. 𝑳𝑹 = 𝟒𝒂 Standard Equation of Parabola Vertex at (0,0) 𝒚𝟐 = 𝟒𝒂𝒙

opens to right

𝒚𝟐 = −𝟒𝒂𝒙

opens to left

𝒙𝟐 = 𝟒𝒂𝒚

opens upward

𝒙𝟐 = −𝟒𝒂𝒚

opens down ward

ELLIPSE ELLIPSE The locus of the point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a. It can also be defined as the locus of the point that moves such that the ratio of its distance from the fixed point, called the focus and the fixed line called the directrix, is constant and less than one (1).

Vertex at (h,k) 𝒚−𝒌

𝟐

= 𝟒𝒂 𝒙 − 𝒉

opens to right

𝒚−𝒌

𝟐

= −𝟒𝒂 𝒙 − 𝒉

opens to left

𝒙−𝒉

𝟐

= 𝟒𝒂 𝒚 − 𝒌

opens upward

𝒙−𝒉

𝟐

= −𝟒𝒂 𝒚 − 𝒌

opens down ward

For the parabola 𝐴𝑥 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 (axis vertical) 𝒉=

−𝑫 𝑫𝟐 − 𝟒𝑨𝑭 −𝑬 ;𝒌 = ;𝒂 = 𝟐𝑨 𝟒𝑨𝑬 𝟒𝑨

Elements of Ellipse 𝒂 𝟐 = 𝒃 𝟐 + 𝒄𝟐 Eccentricity (first eccentricity), 𝒆 =

For the parabola 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 (axis horizontal)

𝒉=

𝑬𝟐

− 𝟒𝑪𝑭 −𝑬 −𝑫 ;𝒌 = ;𝒂 = 𝟒𝑪𝑫 𝟐𝑪 𝟒𝑪

𝒅𝟑 𝒅𝟒

𝒄

= < 𝟏. 𝟎 𝒂

Distance from the center to directrix, 𝒅 = Length of latus rectum, 𝑳𝑹 = Second eccentricity, 𝒆′ = Angular eccentricity, 𝜶 = Ellipse flatness, 𝒇 =

𝒂 𝒆

𝟐𝒃𝟐 𝒂

𝒄 𝒃 𝒄 𝒂

𝒂−𝒃 𝒂

Second flatness, 𝒇 =

𝒂−𝒃 𝒃

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General equation of Ellipse

𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 To solve an ellipse, either one of the following conditions must be known: 1. Four points along the ellipse, Solution: Use the general form 2. Center (h,k), semi-major axis a, and semiminor axis b Solution: Use the standard form

HYPERBOLA HYPERBOLA The locus of the point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a. It can also be defined as the locus of the point that moves such that the ratio of its distance from the fixed point, called the focus and the fixed line called the directrix, is constant and is greater than one (1).

Standard Equations of Ellipse Center at (0,0)

𝒙 𝟐 𝒚𝟐 + =𝟏 𝒂𝟐 𝒃 𝟐

𝒙 𝟐 𝒚𝟐 + =𝟏 𝒃𝟐 𝒂𝟐

Elements of Hyperbola Center at (h,k)

𝒙−𝒉 𝒂𝟐

𝟐

+

𝒄𝟐 = 𝒂 𝟐 + 𝒃 𝟐

𝒚−𝒌 𝒃𝟐

𝟐

=𝟏

Eccentricity 𝒆 =

𝒅𝟑 𝒅𝟒

𝒄

= > 𝟏. 𝟎 𝒂

Distance from the center to directrix, 𝒅 =

𝒂 𝒆

Equation of asymptotes

𝒙−𝒉 𝒃𝟐

𝟐

+

𝒚−𝒌 𝒂𝟐

𝒚 − 𝒌 = ±𝒎 𝒙 − 𝒉

𝟐

=𝟏

Where (h,k) is the center of the hyperbola and m is the slope. Use (+) for upward asymptote and (-) for down ward asymptote.

Note: a>b

𝒎=

For the Ellipse

𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

𝒉=

𝒃 𝒂

𝒎=

𝒂 𝒃

if the axis is horizontal if the axis is vertical

−𝑫 −𝑬 𝒂𝒏𝒅 𝒌 = 𝟐𝑨 𝟐𝑪

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General equation of Hyperbola

𝑨𝒙𝟐 − 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 Standard Equations of Ellipse Center at (0,0)

𝒙𝟐

𝒚𝟐

𝒂

𝒃

− axis =𝟏 Major 𝟐 𝟐 vertical

𝒚𝟐

𝒙𝟐

𝒂

𝒃𝟐

− 𝟐

=𝟏

Major axis horizontal

Major axis vertical

Center at (h,k)

𝒙−𝒉 𝟐 𝒂𝟐

𝒚−𝒌 𝟐 𝒂𝟐

−

+

𝒚−𝒌 𝟐 𝒃𝟐

𝒙−𝒉 𝟐 𝒃𝟐

=𝟏

Major axis horizontal

=𝟏

Major axis vertical

Note: “a” may be greater, equal or less than “b” For the Hyperbola

𝑨𝒙𝟐 − 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

𝒉=

−𝑫 −𝑬 𝒂𝒏𝒅 𝒌 = 𝟐𝑨 𝟐𝑪

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