Question bank / Circles

Math · Geometry and TrigonometryMedium

In the xyxy-plane, the points (−3,4)(-3, 4) and (5,−2)(5, -2) are the endpoints of a diameter of a circle. Which of the following is an equation of the circle?

A

(x−1)2+(y−1)2=25(x - 1)^2 + (y - 1)^2 = 25

B

(x−1)2+(y−1)2=100(x - 1)^2 + (y - 1)^2 = 100

C

(x+1)2+(y+1)2=25(x + 1)^2 + (y + 1)^2 = 25

D

(x−1)2+(y−1)2=5(x - 1)^2 + (y - 1)^2 = 5

Show answer and explanation

Answer: A

The center is the midpoint of the diameter, (−3+52,4+(−2)2)=(1,1)\left(\frac{-3 + 5}{2}, \frac{4 + (-2)}{2}\right) = (1, 1). The length of the diameter is (5−(−3))2+(−2−4)2=64+36=10\sqrt{(5 - (-3))^2 + (-2 - 4)^2} = \sqrt{64 + 36} = 10, so the radius is 55 and r2=25r^2 = 25. An equation of the circle is (x−1)2+(y−1)2=25(x - 1)^2 + (y - 1)^2 = 25.

Choice B uses the diameter, 1010, as the radius, so its right side is 102=10010^2 = 100 instead of 52=255^2 = 25.

Why the other choices are wrong

  • B. This uses the diameter, 10, as the radius.
  • C. The center is (1, 1), so the equation subtracts 1 from x and from y.
  • D. The right side must be the radius squared, 25, not the radius.
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