A circle has a center at P = (–4, 4) . . .

Vincen wrote:A circle has a center at P = (–4, 4) and passes through the point (2, 3). Through which of the following must the circle also pass?

A. (1, 1)
B. (1, 7)
C. (–1, 9)
D. (–3, –2)
E. (–9, 1)

The OA is D.

Should I find the circle equation or only use the distance formula?

The center (–4, 4) is on the line y = –x, so it is equidistant from any point that is a reflection over the line y = –x. The reflection of the given point (2, 3) over the line y = –x is (–3, –2). Thus (–3, –2) must be at the same distance from (–4, 4) as (2, 3) is.

The correct answer: D

Hope this helps!

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Vincen wrote: ↑

Thu Sep 14, 2017 1:50 pm

A circle has a center at P = (–4, 4) and passes through the point (2, 3). Through which of the following must the circle also pass?

A. (1, 1)
B. (1, 7)
C. (–1, 9)
D. (–3, –2)
E. (–9, 1)

The OA is D.

Should I find the circle equation or only use the distance formula?

If we (x, y) is also a point on the circle, then the distance between (x, y) and the center of the circle, (-4, 4), is equal to the distance between (2, 3) and (-4, 4). Therefore, we can create an equation using the distance formula for both sides of the equation as follows:

√[(x - (-4))^2 + (y - 4)^2] = √[(2 - (-4))^2 + (3 - 4)^2]

√[(x + 4)^2 + (y - 4)^2] = √[6^2 + (-1)^2]

(x + 4)^2 + (y - 4)^2 = 37

We see that 37 is an odd number, so one of the squares on the left hand side has to be odd and the other is even. That means one of x and y has to be odd and the other has to be even. From the given answer choices, we see that only choice D satisfies this criterion. Therefore, D is the correct answer.

Answer: D