Coordinate geometry

This chapter covers the following topics:

• Straight line
• Circle
• Sphere
• Parabola
• Ellipse
• Hyperbola
• Conic section equation

Exam tip: All of the formulas in this chapter - circle, sphere, parabola, ellipse, hyperbola, and the general conic section equation - are printed in the “Analytic geometry” part of the mathematics section of the FE Reference Handbook. You don’t need to memorize them for the exam, but you do need to be fast at locating the right one and matching its variables to the numbers in the problem, so practice finding them in the handbook as you work through this chapter.

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:

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

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.,

$$m_1{m}_2=-1$$

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:

$$h=-\frac{D}{2},k=-\frac{E}{2},r=\sqrt{h^2+k^2-F}$$

Example: center and radius from the expanded form

Find the center and radius of the circle $x^2+y^2-6x+4y-12=0$.

Here $D=-6$, $E=4$, $F=-12$, so:

$$h=-\frac{D}{2}=3,k=-\frac{E}{2}=-2,r=\sqrt{h^2+k^2-F}=\sqrt{9+4+12}=5$$

Answer: center $(3,-2)$, radius $5$

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:

$$h=-\frac{D}{2},k=-\frac{E}{2},l=-\frac{F}{2},r=\sqrt{h^2+k^2+l^2-G}$$

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 left/right:

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

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

• Parabola opening up/down:

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

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

Example: finding the vertex, focus, and directrix

Find the vertex, focus, and directrix of the parabola $(y-2{)}^2=8(x-1)$.

This is a left/right-opening parabola with $h=1$, $k=2$, and $4p=8$, so $p=2$.

Answer: vertex $(1,2)$, focus $(1+2,2)=(3,2)$, directrix $x=1-2=-1$

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 $c=\sqrt{a^2-b^2}$
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 $c=\sqrt{a^2-b^2}$
Directrix: $y=k\pm a^2/c$
Eccentricity: $e=c/a$ (where $0<e<1$)

Example: center, foci, and eccentricity of an ellipse

Find the center, foci, and eccentricity of the ellipse $\frac{(x-2{)}^2}{25}+\frac{(y+1{)}^2}{16}=1$.

Since the larger denominator, $25$, sits under the $x$-term, this is a horizontal ellipse with $a^2=25$ (so $a=5$) and $b^2=16$ (so $b=4$). The center is $(h,k)=(2,-1)$.

$$c=\sqrt{a^2-b^2}=\sqrt{25-16}=\sqrt{9}=3$$

Answer: center $(2,-1)$, foci $(2\pm 3,-1)=(5,-1)$ and $(-1,-1)$, eccentricity $e=c/a=3/5=0.6$

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

Example: center, foci, vertices, and asymptotes of a hyperbola

Find the center, foci, vertices, and asymptotes of the hyperbola $\frac{(x-1{)}^2}{9}-\frac{(y-2{)}^2}{16}=1$.

This is a horizontal hyperbola with $h=1$, $k=2$, $a^2=9$ (so $a=3$), and $b^2=16$ (so $b=4$).

$$c=\sqrt{a^2+b^2}=\sqrt{9+16}=\sqrt{25}=5$$

Answer: center $(1,2)$, foci $(1\pm 5,2)=(6,2)$ and $(-4,2)$, vertices $(1\pm 3,2)=(4,2)$ and $(-2,2)$, asymptotes $y-2=\pm \frac{4}{3}(x-1)$

Watch out: For an ellipse, $c=\sqrt{a^2-b^2}$, but for a hyperbola, $c=\sqrt{a^2+b^2}$ - the sign under the square root flips between the two. Also remember that for an ellipse, the larger of $a^2$ and $b^2$ always sits under the variable that runs along the major axis, so check which denominator is bigger before deciding whether the ellipse is horizontal or vertical.

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.

Example: classifying a conic

Classify the conic $4x^2-4xy+y^2+3x-2=0$.

Here $A=4$, $B=-4$, $C=1$, so:

$$B^2-4AC=(-4{)}^2-4(4)(1)=16-16=0$$

Answer: since $B^2-4AC=0$, this equation represents a parabola.