Question 189·200 Super-Hard SAT Math Questions·Geometry and Trigonometry
In the -plane, a circle has its center on the -axis and passes through the points and .
Which choice is an equation of the circle?
When a circle''s center is constrained to an axis or line, build that restriction into the coordinates of the center immediately. Then use equal distances from the center to known points on the circle. This turns the geometry into one solvable algebra equation instead of guesswork.
Hints
Write the center as
Because the center is on the -axis, its -coordinate must be 0.
Set the squared distances equal
The distance from to must equal the distance from to . Use squared distances to avoid square roots.
Desmos Guide
Solve for the y-coordinate of the center
In Desmos, use x as the possible value of k and graph y = 36 + (2 - x)^2 and y = 4 + (10 - x)^2. The x-coordinate of their intersection is the value of k.
Compute r^2 and match the equation
After finding k = 4, enter 36 + (2 - 4)^2. This gives 40, so the matching equation is x^2 + (y - 4)^2 = 40.
Step-by-step Explanation
Model the unknown center
Let the center be . Then the two radii satisfy
Find the center
Expanding gives
So
which leads to
The center is .
Find the equation
Now compute the radius:
So an equation of the circle is