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

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.

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.

:cry: no one is replying
m getting scared of coordinate geometry now

:cry: no one is replying
m getting scared of coordinate geometry now

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

@dave13 : (37 is what ? ) distance ? distance between which points
37 is the length of the radius, squared.
The radius is the distance between the center of a circle and any point that lies on the circle.
(The length of the radius is \(\sqrt{37}\))

where from did you get various values that you plugged in answer choices?
I wrote the equation for THIS circle first (from the "standard" formula you know).
Then I found the radius using the given point (2,3).
(Try plugging that point into the equation for THIS circle in blue, directly below.)

The equation for THIS circle is \((x + 4)^2 + (y - 4)^2 = r^2\)
Then, in the answers, values I "plugged in" were x- and y-coordinates for the points in the answers.
I just did not show all the steps. I show full steps below for Answers D and A.

You say you know this formula: \((x−a)^2\)+\((y−b)^2\) =\(r^2\)
Excellent. Standard Equation of a circle,* which is used for circles whose centers are NOT at (0,0)
The center is (a,b). The radius is r. (I used the same equation with (h, k).)
That is the formula I used to test the answers.
But you have to modify the Standard Formula to fit this particular circle.

\((x−a)^2\)+\((y−b)^2\) =\(r^2\)
You need a center (given), to find the equation for this particular circle.
You need one point on the circle (given) to find the length of the radius, squared.

Find equation for THIS circle. Plug the center's coordinates (-4,4) into standard circle equation:
\((x - (-4))^2 + (y - 4)^2 = r^2\)
\((x + 4)^2 + (y - 4)^2 = r^2\) - equation for THIS circle

Find the radius.
Any point that lies on the circle will satisfy this equation (will make the equation true).
That point's coordinates will yield the length of the radius, squared.
It is given that (2,3) lies on the circle. Plug in point (2,3):
\((2 + 4)^2 + (3 - 4)^2 = r^2\)
\((6)^2 + (-1)^2 = r^2\)
\(36 + 1 = 37 = r^2\) OR
\(r = \sqrt{37}\)

The way I solved this problem, you can and should keep the radius's length squared. Why?
Because any other point that lies on the circle has to satisfy the equation, and --
we're squaring everything! The equation gives us \(r^2\).
Why make more work?

To find whether a point is on the circle, I have to plug (x,y) into the equation we found above:
\((x + 4)^2 + (y - 4)^2 = r^2\)
Any point on the circle, when I plug its coordinates into the equation, will yield \(37 = r^2\)

Let's take the other point on the circle, Answer D: (-3, -2). Plug in its coordinates.
The result MUST be \(37 = r^2\)
\((x + 4)^2 + (y - 4)^2 = r^2\)
\((-3 + 4)^2 + (-2 - 4)^2 = r^2\)
\((-1)^2 + (-6)^2 = r^2\)
\(1 + 36 = 37 = r^2\) BINGO
If you want to take the extra step, \(\sqrt{r^2}\), that is fine.

(That step is often used in the distance formula. I'm not a fan of the distance formula.)

Do any of the OTHER answers work? (NO.)
\((x + 4)^2 + (y - 4)^2 = r^2\). Plug in x and y. Result must be 37

• Answer A) point (1,1)?

\((1 + 4)^2 + (1 - 4)^2 = r^2\)
\((5)^2 + (-3)^2 = r^2\)
\(25 + 9 = r^2\)
\(25 + 9 = 34 = r^2\)
\(r^2\) should equal 37. Here, \(r^2\) equals 34.

NO, this point does not lie on the circle.
WRONG RADIUS LENGTH
Length of radius for this point? \(r = \sqrt{34}\)
True length of radius for the circle? \(r = \sqrt{37}\)

Try the other answers. Plug the x- and y-coordinates into the equation.
Do they result in \(r^2 = 37\) or (longer way) \(r = \sqrt{37}\)?

Many thanks generis for unique explanation one question why did you reduce the font size of the important part of explanation
its like in contract where the font size is very small of the most important part of contract

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.