In the xy-plane, circle C is tangent with x-axis and y-axis.

From (1) -> Area = pi 4^2 so r = 4, so the center can be (4,4), (-4,-4), (-4,4), (4,-4), but nothing about m or n, so not sufficient

From (2) no information about n, so not sufficient

From (1) and (2) (m,n) can be (8,4) or (8,-4) so answer is E

can you please tell me how we can find n=4 or -4? i dont understand this part

awesome post. but why are you assuming that the point lies along the daimeter (\(n\) is either 4 or -4) am I missing sth?

From both statements; we know that x=8; x=8 is true only for points (8,4) and (8,-4) {the extreme right point of the circles}. In the figure drawn by Bunuel, please see the red line parallel to the y-axis and watch where it's intersecting the circles.

Another way to find the point would be to use the equation of the circle:

For a circle centered at (a,b) and radius 'r', the equation of the circle would be:
\((x-a)^2+(y-b)^2=r^2\)

Here, the first circle has the center at (4,4) and the radius is 4
Thus, equation of the circle will be:

\((x-4)^2+(y-4)^2=4^2\)
We know x=8
\((8-4)^2+(y-4)^2=4^2\)
\(4^2+(y-4)^2=4^2\)
\((y-4)^2=0\)
\(y=4\)

the second circle has the center (4,-4) and the radius is 4
Thus, equation of the circle will be:

\((x-4)^2+(y-(-4))^2=4^2\)
We know x=8
\((8-4)^2+(y+4)^2=4^2\)
\(4^2+(y+4)^2=4^2\)
\((y+4)^2=0\)
\(y=-4\)

Hence two values of y for x=8.

Bunuel, awesome explanation!

Given: Circle C is tangent with x-axis and y-axis. Point (m,n) lies on the circle
Required: m + n

Statement 1: The area of the circle is 16pi
Area of a circle = 2*pi*r^2
Hence we have radius = 4.
This does not give any information about m and n
INSUFFICIENT

Statement 2: m = 8
Here again, we do not have any other information to take out the value of n,
INSUFFICIENT

Combining Statement 1 and Statement 2:
Area = 16pi and m = 8

Hence we can have two circles above and below the X axis.
Thus, we can have two values of n (one positive and one negative)
Still no single answer

INSUFFICIENT

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In the xy-plane, circle C is tangent with x-axis and y-axis. If point (m, n) lies on the circle, what is the value of m + n?

(1) The area of the circle is 16pi.
(2) m = 8.

We have a circle with the center being (r,-r), (-r,r),(-r,-r),(r,-r), and the radius r. We need to know the center, so there are 2 variables (r, the quadrant); 2 equations are given from the 2 conditions, so there is high chance (C) will be our answer.
Looking at the conditions together, the radius becomes 4, so m^2+n^2=4^2=16, and if we substitute in m=8, there is no value of n that satisfies the conditions, which means the question is not really appropriate... it is (E) if we really must choose an answer.

if to picture the circle...it can be in the first quadrant, in the second quadrant, third, and fourth
so basically, we need to know where it is situated...

1. r=4, but it can be in any quadrant
2. m=8, so it can be in first or fourth quadrant

1+2 - it can be either in first or in fourth quadrant.

E

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