Conic sections

A conic section is the shape you get when you slice through a cone. For the ACT or SAT, you only need to recognize parabolic, circular, and elliptical cross sections. They look like this:

In this chapter, you won’t focus on how these cross sections are formed. Instead, you’ll focus on how to work with their equations.

You may already know how to solve parabolic equations. If not, read the chapter Solving quadratic equations before continuing, since we’ll only discuss circular cross sections in further detail.

Circular cross-sections

A circle can be described by an equation, just like a parabola. The standard form of a circle’s equation is:

$$(x-a{)}^2+(y-b{)}^2=r^2$$

If the circle is centered at the origin, then $a=0$ and $b=0$, so the equation simplifies to:

$$x^2+y^2=r^2$$

To use these equations correctly, you need to understand what each part represents.

Equation of a circle: center

In the equation of a circle:

• $x$ and $y$ are the coordinate directions.
• $r$ is the radius.
• $(a,b)$ is the center of the circle.

That’s why the origin-centered version has no $a$ or $b$: both are $0$.

In most problems, you’ll read the center directly from the equation:

• $a$ is the $x$-coordinate of the center.
• $b$ is the $y$-coordinate of the center.

Example: Find the center of the circle given by

$$(x-4{)}^2+(y+3{)}^2=36$$

Here, $a=4$ and $b=-3$, so the center is $(4,-3)$.

Equation of a circle: radius

A common mistake is forgetting that the radius is squared in the equation. The number on the right side is $r^2$, not $r$.

So to find the radius, take the square root of the right-hand side.

Referring to the example above, $(x-4{)}^2+(y+3{)}^2=36$, what is the radius of the circle?

Here’s another example where you use the radius to find something else.

A circle is described by the following equation:

$$(x-1{)}^2+(y-1{)}^2=144$$

What is the circumference of this circle?

To find the circumference, you first need the radius.

A similar idea shows up with parabolas. Quadratic functions can appear in vertex form, which gives the vertex directly:

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

The vertex is $(h,k)$. More information about this form is in the chapter “Vertex form equation ($a$, $h$, and $k$)”.

Elliptical cross sections

An ellipse is essentially a “stretched” circle: it can be wider or taller than a perfect circle. You should know the equation of an ellipse and what each term means.

The equation tells you how far the ellipse extends from its center in each direction:

• $a$ is the distance from the center to the ellipse in the $x$-direction.
• $b$ is the distance from the center to the ellipse in the $y$-direction.
• $(h,k)$ is the center.

Here is the equation of an ellipse:

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

Remember:

• $a$ is paired with the $x$-term, so it controls the horizontal stretch.
• $b$ is paired with the $y$-term, so it controls the vertical stretch.

That means:

• If $a>b$, the ellipse is wider than it is tall.
• If $a<b$, the ellipse is taller than it is wide.
• If $a=b$, the ellipse is actually a circle.

Visualization

To visualize the values of $a$ and $b$, see the following image: