Analytic Geometry Formulas

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CARTESIAN/RECTANGULAR COORDINATE SYSTEM

DISTANCE BETWEEN TWO POINTS

𝒅 = 𝒙𝟐− 𝒙𝟏 𝟐+ 𝒚𝟐− 𝒚𝟏 𝟐

The distance between two point 𝑃 𝑥1, 𝑦1 and 𝑄 𝑥2, 𝑦2 is: 𝒚 − 𝒚𝟏= 𝒎 𝒙 − 𝒙𝟏 𝒚 = 𝒎𝒙 + 𝒃 𝒙 𝒂+ 𝒚 𝒃= 𝟏 𝒚 − 𝒚𝟏 𝒙 − 𝒙𝟏 = 𝒚𝟐− 𝒚𝟏 𝒙𝟐− 𝒙𝟏

STANDARD EQUATION OF LINES

1. Point-Slope form

Given a point 𝑃1 𝑥1, 𝑦1 and slope 𝑚.

2. Slope-Intercept form

Given a slope 𝑚 and 𝑦-intercept 𝑏:

3. Intercept form

Given x-intercept 𝑎 and y-intercept 𝑏:

4. Two-point form

Given two points 𝑃1 𝑥1, 𝑦1 and 𝑃2 𝑥2, 𝑦2 :

SLOPE OF THE LINE

𝒔𝒍𝒐𝒑𝒆, 𝒎 =𝒓𝒊𝒔𝒆 𝒓𝒖𝒏=

𝒚𝟐− 𝒚𝟏 𝒙𝟐− 𝒙𝟏

The slope of the line passing through points 𝑃 𝑥1, 𝑦1 and

𝑄 𝑥2, 𝑦2 is:

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

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0

GENERAL EQUATION OF A LINE

The general equation of a straight line is:

EQUATION OF A LINE

𝐭𝐚𝐧 𝜽 = 𝒎𝟐− 𝒎𝟏 𝟏 + 𝒎𝟏𝒎𝟐

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

Lines are parallel if 𝑚1= 𝑚2

Lines are perpendicular if 𝑚2= _𝑚−1

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DISTANCE FROM A POINT TO A LINE

𝒅 =𝑨𝒙𝟏+ 𝑩𝒚𝟏+ 𝑪 ± 𝑨𝟐_{+ 𝑩}𝟐

The distance (nearest) from a point 𝑃1 𝑥1, 𝑦1 to a line

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 is:

DISTANCE BETWEEN TWO PARALLEL LINES

𝒅 = 𝑪𝟐− 𝑪𝟏 𝑨𝟐_{+ 𝑩}𝟐

The distance between two parallel lines 𝐿1∶ 𝐴𝑥 + 𝐵𝑦 + 𝐶1

and 𝐿2∶ 𝐴𝑥 + 𝐵𝑦 + 𝐶2 is:

𝒙

=

𝒓𝟏+𝒓𝟐

𝒚

=

𝒚𝟏𝒓𝟐+𝒚𝟐𝒓𝟏

𝒓𝟏+𝒓𝟐

DIVISION OF LINE SEGMENT

MIDPOINT OF A LINE SEGMENT

𝒙𝒎=

𝒙𝟏+ 𝒙𝟐

𝟐 𝒚𝒎=

𝒚𝟏+ 𝒚𝟐 𝟐

The midpoint 𝑃𝑚 𝑥𝑚, 𝑦𝑚 of a line segment

through from 𝑃1 𝑥1, 𝑦1 to 𝑃2 𝑥2, 𝑦2 is:

CONIC SECTIONS

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.

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.

GENERAL EQUATION OF CONICS

𝑨𝒙𝟐_{+ 𝑪𝒚}𝟐_{+ 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

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GENERAL EQUATION OF CONICS

𝑨𝒙𝟐_{+ 𝑩𝒙𝒚 + 𝑪𝒚}𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎} 𝑨𝒙𝟐_{+ 𝑪𝒚}𝟐_{+ 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.

r = radius (h,k) = center

General equation of a Circle (A=C) or

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

Center at (h,k) 𝒙 − 𝒉 𝟐_{+ 𝒚 − 𝒌} 𝟐 _{= 𝒓}𝟐

Center at (0,0) 𝒙𝟐_{+ 𝒚}𝟐_{= 𝒓}𝟐

For the circle

PARABOLA

𝒙𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎}

𝑪𝒚𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎} 𝒚𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎}

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.

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

General equation of Parabola (A or C is zero) C = 0

𝑨𝒙𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 0} or

A = 0 or

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

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Latus rectum is the chord passing through the focus and parallel to directrix or perpendicular to the axis.

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}

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 𝐴𝑥2_{+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0}

(axis vertical)

For the parabola 𝐶𝑦2_{+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0}

(axis horizontal)

ELLIPSE

𝒂𝟐_{= 𝒃}𝟐_{+ 𝒄}𝟐

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).

Elements of Ellipse

Eccentricity (first eccentricity), 𝒆 =𝒅𝟑

𝒅𝟒=

𝒄 𝒂< 𝟏. 𝟎

Distance from the center to directrix, 𝒅 =𝒂 𝒆

Length of latus rectum,𝑳𝑹 =𝟐𝒃_𝒂𝟐

Second eccentricity, 𝒆′₌𝒄 𝒃 Angular eccentricity, 𝜶 =𝒄_𝒂 Ellipse flatness, 𝒇 =𝒂−𝒃 𝒂 Second flatness, 𝒇 =𝒂−𝒃_𝒃

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HYPERBOLA

𝒙

𝒂

+

𝒚

𝒃

= 𝟏

𝒙

𝒃

+

𝒚

𝒂

= 𝟏

General equation of Ellipse

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), major axis a, and semi-minor axis b

Solution: Use the standard form

Standard Equations of Ellipse Center at (0,0)

Center at (h,k)

Note: a>b For the Ellipse

𝒄𝟐_{= 𝒂}𝟐_{+ 𝒃}𝟐

𝒚 − 𝒌 = ±𝒎 𝒙 − 𝒉

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). Elements of Hyperbola Eccentricity 𝒆 =𝒅𝟑 𝒅𝟒= 𝒄 𝒂> 𝟏. 𝟎

Distance from the center to directrix, 𝒅 =𝒂_𝒆

Equation of asymptotes

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

𝒎 =𝒃_𝒂 if the axis is horizontal

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Major axis vertical

𝑨𝒙𝟐_{− 𝑪𝒚}𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎} 𝑨𝒙𝟐_{− 𝑪𝒚}𝟐_{+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎} 𝒉 =−𝑫 𝟐𝑨 𝒂𝒏𝒅 𝒌 = −𝑬 𝟐𝑪

General equation of Hyperbola

Standard Equations of Ellipse Center at (0,0)

𝒙𝟐

𝒂𝟐

−

𝒚𝟐

𝒃𝟐

= 𝟏

𝒚𝟐

𝒂𝟐

−

𝒙𝟐

𝒃𝟐

= 𝟏

Center at (h,k) 𝒙−𝒉 𝟐

𝒂𝟐

−

𝒚−𝒌 𝟐

𝒃𝟐

= 𝟏

𝒚−𝒌 𝟐

𝒂𝟐

+

𝒙−𝒉 𝟐

𝒃𝟐

= 𝟏

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