SAT Circles Question 18 | Free SAT Prep

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Question 18·Hard·Circles

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In the $xy$ -plane, circle $C$ is defined by the equation

x^{2}+y^{2}-6x+8y+9=0.

The line $\ell$ has equation $y=-x-1$ and intersects circle $C$ at distinct points $P$ and $Q$ . What is the length of $\overline{PQ}$ ?

For circle-and-line intersection questions that ask for a chord length, first rewrite the circle in standard form $(x - h)^2 + (y - k)^2 = r^2$ to quickly read off the center and radius. Then compare the line to the center—either plug the center into the line or use the distance-from-point-to-line formula—to see whether the line passes through the center. If it does, the intersection segment is a diameter, so you can immediately set its length to $2r$ without solving the full system of equations, which saves time and avoids messy algebra.

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Put the circle into a more useful form

Try rewriting the circle's equation in the form $(x - h)^2 + (y - k)^2 = r^2$ by completing the square for both $x$ and $y$ . This will give you the center and radius.

Relate the line to the circle's center

Once you have the center of the circle, substitute its coordinates into the line's equation. Does the point satisfy the line's equation?

Interpret the geometry

If a line passes through the center of a circle and intersects the circle in two points, what is the segment between those two points called, and how is its length related to the radius?

Connect radius and the segment length

Use the relationship between radius and diameter to express the length of $\overline{PQ}$ in terms of the radius you found.

Desmos Guide

Graph the circle

In Desmos, enter the circle equation exactly as given: x^2 + y^2 - 6x + 8y + 9 = 0. Desmos will graph the circle.

Graph the line

Enter the line equation y = -x - 1. You should see the line passing through and cutting across the circle in two places.

Find the intersection points

Click on each point where the line and circle intersect. Desmos will label them (for example, A and B) and show their coordinates.

Use Desmos to measure the distance

In a new expression line, type distance(A, B) (using whatever labels Desmos gave the two intersection points). The value that Desmos outputs is the length of $\overline{PQ}$ .

Step-by-step Explanation

Rewrite the circle in standard form

Start with the equation of circle $C$ :

x^2 + y^2 - 6x + 8y + 9 = 0

Group the $x$ -terms and $y$ -terms and complete the square:

(x^2 - 6x) + (y^2 + 8y) + 9 = 0

Complete the square for each group:

For $x^2 - 6x$ , add and subtract $9$ .
For $y^2 + 8y$ , add and subtract $16$ .

So we get

(x^2 - 6x + 9) + (y^2 + 8y + 16) + 9 - 9 - 16 = 0

which simplifies to

(x - 3)^2 + (y + 4)^2 - 16 = 0

(x - 3)^2 + (y + 4)^2 = 16

The center of the circle is $(3,-4)$ and the radius is $r = 4$ .

Check whether the line passes through the center

The line is given by $y = -x - 1$ . Rewrite it in standard form:

y = -x - 1 \Rightarrow x + y + 1 = 0

Now plug the center $(3,-4)$ into $x + y + 1$ :

3 + (-4) + 1 = 3 - 4 + 1 = 0

Since the expression equals $0$ , the center $(3,-4)$ lies on the line $x + y + 1 = 0$ , so line $\ell$ passes through the center of the circle.

Use the geometry of a circle and a line through its center

If a line passes through the center of a circle and intersects the circle at two distinct points $P$ and $Q$ , then $\overline{PQ}$ is a diameter of the circle.

The length of a diameter is related to the radius by

\text{diameter} = 2r

So here, $PQ$ must have length $2r$ .

Compute the diameter length

From Step 1, the radius of the circle is $r = 4$ .

So the length of $\overline{PQ}$ is

PQ = 2r = 2 \cdot 4 = 8

Therefore, the correct answer is $8$ (choice C).

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation