Question 1163739
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Use "^" to denote exponents: x^2+y^2-12x-4y=50....<br>
Complete the square in x and y to find the center and radius.<br>
{{{(x^2-12x+36)+(y^2-4y+4) = 50+36+4 = 90}}}<br>
{{{(x-6)^2+(y-2)^2 = 90}}}<br>
The center is at (6,2); the radius is {{{sqrt(90)}}}.<br>
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.<br>
The slope of the line through (0,0) and (6,2) is 1/3.  So we need to move a distance of {{{sqrt(90)}}} away from (6,2) along the line {{{y = (1/3)x}}}.<br>
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.<br>
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).<br>