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

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

by Vincen » 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?
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by Jay@ManhattanReview » Thu Sep 14, 2017 10:23 pm
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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Re: A circle has a center at P = (–4, 4) . . .

by Scott@TargetTestPrep » Sun Mar 29, 2020 4:12 am
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

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