Coordinate geometry

This chapter covers the following topics:

• Straight Line
• Circle
• Sphere
• Parabola
• Ellipse
• Hyperbola
• Conic Section Equation

Straight line

General form

The general form of a straight-line equation is $Ax+By+C=0$, where $A,B,C$ are real constants, and $A$ and $B$ are not both zero.

Standard form

The standard form of a straight line is $Ax+By=C$, where $A,B,C$ are integers, and $A$ is usually positive.

Slope (m)

The slope of a straight line tells you how fast $y$ changes as $x$ changes. It’s defined by:

$$m=\frac{\Delta y}{\Delta x}=\frac{(y_2-y_1)}{(x_2-x_1)}$$

where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

Intercepts

X-intercept: The point where the line crosses the x-axis $(y=0)$. From the general form, $x=-C/A$.

Y-intercept: The point where the line crosses the y-axis $(x=0)$. From the general form, $y=-C/B$.

Slope-intercept form

If a line has slope $m$ and y-intercept $b$, you can write it as:

$$y=mx+b$$

where $m$ is the slope and $b$ is the y-intercept.

Point-slope form

If a line passes through a known point $(x_1,y_1)$ and has slope $m$, its equation is:

$$y-y_1=m(x-x_1)$$

Two-point form

If a line passes through two given points $(x_1,y_1)$ and $(x_2,y_2)$, its equation can be written as:

$$y-y_1=\frac{(y_2-y_1)}{(x_2-x_1)}\ast (x-x_1)$$

Conditions for parallel and perpendicular lines

Parallel lines: Two lines are parallel if and only if they have the same slope $(m_1=m_2)$.

Perpendicular lines: Two lines are perpendicular if and only if the product of their slopes is $-1$, i.e.,

$m_1\ast m_2=-1$

The distance between two points is:

$$d=\sqrt{(}(y_2-y_1{)}^2+(x_2-x_1{)}^2)$$

Circle

Standard form of the equation of a circle

$$(x-h{)}^2+(y-k{)}^2=r^2$$

where $(h,k)$ is the center of the circle, and $r$ is the radius.

Expanded form of a circle

$$x^2+y^2+Dx+Ey+F=0$$

where $D,E,F$ are constants. The center and radius are:

$$h=-D/2,k=-E/2,r=\surd (h^2+k^2-F)$$

Sphere

Standard form of the equation of a sphere

$$(x-h{)}^2+(y-k{)}^2+(z-l{)}^2=r^2$$

where $(h,k,l)$ is the center and $r$ is the radius.

Expanded form of a sphere

$$x^2+y^2+z^2+Dx+Ey+Fz+G=0$$

where $D,E,F,G$ are constants. The center and radius are:

$$h=-D/2,k=-E/2,l=-F/2,r=\surd (h^2+k^2+l^2-G)$$

Parabola

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Standard form of a parabola

• Parabola opening up/down:

$$(y-k{)}^2=4p(x-h)$$

Vertex: $(h,k)$
Focus: $(h+p,k)$
Directrix: $x=h-p$
Axis of symmetry: $y=k$

• Parabola opening left/right:

$$(x-h{)}^2=4p(y-k)$$

Vertex: $(h,k)$
Focus: $(h,k+p)$
Directrix: $y=k-p$
Axis of symmetry: $x=h$

Ellipse

An ellipse is the set of all points where the sum of the distances to two fixed points (the foci) is constant.

Standard form of an ellipse

• Horizontal ellipse:

$$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$$

Center: $(h,k)$
Major axis length: $2a$
Minor axis length: $2b$
Foci: $(h\pm c,k)$, where $c=\surd (a^2-b^2)$
Directrix: $x=h\pm a^2/c$
Eccentricity: $e=c/a$ (where $0<e<1$)

• Vertical ellipse:

$$\frac{(x-h{)}^2}{b^2}+\frac{(y-k{)}^2}{a^2}=1$$

Center: $(h,k)$
Major axis length: $2a$
Minor axis length: $2b$
Foci: $(h,k\pm c)$, where $c=\surd (a^2-b^2)$
Directrix: $y=k\pm a^2/c$
Eccentricity: $e=c/a$ (where $0<e<1$)

Hyperbola

A hyperbola is the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant.

Standard form of a hyperbola

• Horizontal hyperbola:

$$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$$

Center: $(h,k)$
Foci: $(h\pm c,k)$, where $c=\surd (a^2+b^2)$
Vertices: $(h\pm a,k)$
Directrix: $x=h\pm a^2/c$
Eccentricity: $e=c/a$ (where $e>1$)
Asymptotes: $y-k=\pm b/a(x-h)$

• Vertical hyperbola:

$$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$$

Center: $(h,k)$
Foci: $(h,k\pm c)$, where $c=\surd (a^2+b^2)$
Vertices: $(h,k\pm a)$
Directrix: $y=k\pm a^2/c$
Eccentricity: $e=c/a$ (where $e>1$)
Asymptotes: $y-k=\pm a/b(x-h)$

Conic section equation

The general form of a conic section equation is:

$$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$$

where:

• If $B^2-4AC<0$, the conic is an ellipse.
• If $B^2-4AC>0$, the conic is a hyperbola.
• If $B^2-4AC=0$, the conic is a parabola.
• If $A=C$ and $B=0$, the conic is a circle.
• If $A=B=C=0$, the equation represents a straight line.