Super-Hard SAT Geometry Question 22 | Free SAT Prep

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Question 22·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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In the $xy$ -plane, circle $T$ has its center on the line $y=x$ and contains the points $(0,6)$ and $(8,0)$ . Circle $U$ is obtained by reflecting circle $T$ across the $x$ -axis and then doubling its radius. The equation of circle $U$ can be written as

(x-a)^2+(y-b)^2=p,

where $a$ , $b$ , and $p$ are constants. What is the value of $p$ ?

Your answer

When a circle’s center is restricted to a line, represent the center using one variable based on that line. Then use the fact that every point on the circle is the same distance from the center to solve for the center. Finally, remember that the right side of a standard circle equation is $r^2$ , so changes to the radius must be squared.

Hints

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Use the line containing the center

Any point on the line $y=x$ can be written as $(t,t)$ .

Use equal radii

The center is the same distance from $(0,6)$ as from $(8,0)$ . Set the two squared-distance expressions equal.

Connect radius to the equation

In a circle equation of the form $(x-h)^2+(y-k)^2=r^2$ , the constant on the right is the square of the radius. Consider how that constant changes when the radius is doubled.

Desmos Guide

Graph the center conditions

Algebra is faster for this problem, but Desmos can verify the center. Graph $y=x$ and

x^2+(y-6)^2=(x-8)^2+y^2.

The intersection represents the center of circle $T$ .

Calculate the original squared radius

Click the intersection to identify the center as $(7,7)$ . Then enter

7^2+(7-6)^2

to find the squared radius of circle $T$ .

Scale the squared radius

Enter $4(50)$ because doubling a radius multiplies its square by $4$ . The displayed value confirms that $p=200$ .

Step-by-step Explanation

Represent the center

Because the center of circle $T$ lies on $y=x$ , write its coordinates as $(t,t)$ . Since both given points are on circle $T$ , their distances from the center are equal:

t^2+(t-6)^2=(t-8)^2+t^2.

Solving gives $t=7$ , so the center of circle $T$ is $(7,7)$ .

Find the squared radius of circle $T$

Use the point $(0,6)$ and the center $(7,7)$ :

r_T^2=(7-0)^2+(7-6)^2=49+1=50.

Reflecting a circle does not change its radius.

Apply the radius change

Circle $U$ has radius $2r_T$ , so its squared radius is

(2r_T)^2=4r_T^2=4(50).

Therefore, $p=200$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation