In the xy-plane above, the circle has center (h, k) and radius 10. What is the value of k ?

Answer

6

Explanation

Circles are symmetric with respect to any given diameter through the center (h, k). One diameter of the circle is perpendicular to the x-axis. Therefore, the value of h is the mean of the x-coordinates of the circle’s two x-intercepts: h = `(20+4)/2=12`. The radius of the circle is given as 10, so the point (h, k) must be a distance of 10 units from any point on the circle. The equation of any circle can be written as `(x − h)^2 + (y − k)^2 = r^2`, where (h, k) is the center of the circle and r is the length of the radius of the circle. Substituting 12 for h and 10 for r into this equation gives `(x − 12)^2 + (y − k)^2 = 10^2`. Substituting the x-coordinate and y-coordinate of a point on the circle, (4, 0), gives `(4 − 12)^2 + (0 − k)^2 = 10^2`, or `64 + k^2 = 100`. Subtracting 64 from both sides of this equation yields `k^2` = 36. Therefore, `k = pm sqrt36` . Since the graph shows the point (h, k) in the first quadrant, k must be the positive square root of 36, so k = 6.