A circle C has center at the origin and radius 9. Another circle K has a diameter with one end at the origin and the other end at the point (0,17). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let (r,theta) be the polar coordinates of P, chosen so that r is positive and 0<theta<2 inclusive. Find r and theta.
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To find the polar coordinates (r, θ) of point P, we can use the coordinates of the intersection points of the circle C and K.
Let's first find the intersection points:
Circle C has its center at the origin (0, 0) and radius 9. In polar coordinates, this can be written as C: r = 9.
Circle K has a diameter with one end at the origin (0, 0) and the other end at the point (0, 17). The radius of K is half the length of its diameter, which is 17/2. In polar coordinates, this can be written as K: r = 17/2.
Now, we have the equations:
C: r = 9
K: r = 17/2
To find the intersection points, we equate the r values:
9 = 17/2
Simplifying, we get:
18 = 17
This equation is not true, which means the circles C and K do not intersect. Therefore, there are no intersection points P, and we cannot find the polar coordinates (r, θ) for P in this scenario.