SOLUTION: Which point on the circle x2+y2−12x−4y= 50 is closest to the origin? Which point is farthest from the origin?

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Question 1163739: Which point on the circle x2+y2−12x−4y= 50 is closest to the origin? Which point is farthest from the origin?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39620) About Me  (Show Source):
Answer by greenestamps(13203) About Me  (Show Source):
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Use "^" to denote exponents: x^2+y^2-12x-4y=50....

Complete the square in x and y to find the center and radius.

%28x%5E2-12x%2B36%29%2B%28y%5E2-4y%2B4%29+=+50%2B36%2B4+=+90

%28x-6%29%5E2%2B%28y-2%29%5E2+=+90

The center is at (6,2); the radius is sqrt%2890%29.

The points on the circle that are closest to and farthest away from the origin are the two endpoints of the diameter of the circle that passes through the origin.

The slope of the line through (0,0) and (6,2) is 1/3. So we need to move a distance of sqrt%2890%29 away from (6,2) along the line y+=+%281%2F3%29x.

Various calculations (or insight) tell us we need to move 9 units in the x direction and 3 units in the y direction from the center of the circle to get to the two points we are looking for.

ANSWERS:
The point on the circle closest to the origin is (6-9,2-3) = (-3,-1).
The point on the circle farthest from the origin is (6+9, 2+3) = (15,5).