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A circle has a center at P = (–4, 4) and passes through the point (2,

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A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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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)


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[Reveal] Spoiler: OA

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Re: A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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radius = distance between center and point = \(\sqrt{(2-(-4))^2+(3-4)^2}\) =\(\sqrt{37}\)

use this distance formula on all the the options they should have same distance. option D. (–3, –2) has the same distance Answer is D.

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Re: A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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Bunuel 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)


Kudos for a correct solution.


Since this is a circle, the distance between any point and the radius will always be the same.
Equation of circle is given by formula, (x-h)^2 + (y-k)^2 = r^2 where (h,k) is center of circle.
Hence r^2 = (2+4)^ 2+ (3 - 4)^2
So, r^2 = 37
So, r = sqrt(37)

This should be valid for any other point as well.
Since we use terms like (x-h)^2 and (y-k)^2 i.e. squaring the expression, let us first try the option D because it contains the same numerical values as the point given in the question.
So, r^2 = (-3+4)^ 2+ (-2 - 4)^2
= 37
Hence this is the point on circle.
Hence option D.

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Re: A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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Bunuel 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)


Kudos for a correct solution.


MAGOOSH OFFICIAL SOLUTION:

For this problem, there’s a long tedious way to slog through the problem, and there’s a slick elegant method that gets to the answer in a lightning fast manner.
The long slogging approach — first, calculate the distance from (–4, 4) to (2, 3). As it happens, that distance, the radius, equals \(\sqrt{37}\) . Then, we have to calculate the distance from (–4, 4) to each of the five answer choices, and find which one has also has a distance of \(\sqrt{37}\) —- all without a calculator. :(

The slick elegant approach is as follows. The point (–4, 4) is on the line y = –x, so it is equidistant from any point and that point’s reflection over the line y = –x. The reflection of (2, 3) over the line y = –x is (–3, –2). Since (–3, –2) is the same distance from (–4, 4) as is (2, 3), it must also be on the circle.

Answer = D.
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Resources:
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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New post 28 Aug 2017, 12:44
The point (–4, 4) is on the line y = –x, so it is equidistant from any point and that point’s reflection over the line y = –x. The reflection of (2, 3) over the line y = –x is (–3, –2). Since (–3, –2) is the same distance from (–4, 4) as is (2, 3), it must also be on the circle. Answer = D.

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A circle has a center at P = (–4, 4) and passes through the point (2, [#permalink]

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New post 30 Aug 2017, 07:58
Bunuel 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)

Kudos for a correct solution.

I didn't catch shortcut y = -x, which IS elegant. I'm not a fan of the distance formula.

Start with general formula for a circle:

\((x - h)^2 + (y - k)^2 = r^2\). Plug in center coordinates:

\((x + 4)^2 + (y - 4)^2 = r^2\). Any point through which the circle passes will give you the radius squared. Plug in given point (2,3)

\((2 + 4)^2 + (3 - 4)^2 = r^2\)
\(6^2 + (-1)^2 = r^2\)
\(36 + 1 = 37 = r^2\)

Don't solve for \(r\) in this problem with this method. You don't need it.

In the answer choices, we need x and y coordinates, that, when plugged into the equation, sum to 37 (satisfy the equation). The circle will pass through that point.

Scanning answer choices, because equation contains +4 and -4 and squares, look for values similar to (2,3) and test them [Answer D's (-3,-2)]. Or do the math for each choice, which is quick when using the circle equation.

A. (1,1): \(5^2 + (-3)^2 = 34\) NO

B. (1,7): \((-3)^2 + 4^2 = 25\) NO

C. (–1, 9): \((-3)^2 + 5^2 = 34\) NO

D. (–3, –2): \((-1)^2 + (-6)^2 = 37\) YES

E. (–9, 1): \((-5)^2 + (-3)^2 = 34\) NO

Answer D

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A circle has a center at P = (–4, 4) and passes through the point (2,   [#permalink] 30 Aug 2017, 07:58
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