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.

Triangle with two sides of length 8.5 and one of length 9

side 1 origin to (0,8.5 ), length 8.5
side 2, (0,8.5) to P, length 8.5
side 3, origin to P, length 9 (this is r of course)

find angle in triangle at origin
cos A = 4.5/8.5
find A from that
then
Theta = 90-A in degrees or pi/2 - A if using radians

To find the polar coordinates (r, theta) of the point P, we need to determine the values of r (radius) and theta (angle).

Let's start by determining the equation of both circles:

Circle C:
Center = (0, 0)
Radius = 9

Circle K:
Center = (0, 17/2) (midpoint of the diameter)
Diameter = distance between (0, 0) and (0, 17) = √((0-0)^2 + (0-17)^2) = √289 = 17
Radius = Diameter / 2 = 17 / 2 = 8.5

The equation of a circle centered at the origin is given by r^2 = x^2 + y^2.

For circle C: r^2 = x^2 + y^2 = 9^2 = 81
For circle K: r^2 = x^2 + (y - 17/2)^2 = 8.5^2 = 72.25

To find the points of intersection, we need to solve these two equations simultaneously:

81 = x^2 + y^2
72.25 = x^2 + (y - 17/2)^2

Simplifying the second equation:
72.25 = x^2 + y^2 - 17y + (17/2)^2
72.25 = x^2 + y^2 - 17y + 289/4
0 = x^2 + y^2 - 17y + (289/4 - 72.25)
0 = x^2 + y^2 - 17y + 289/4 - 289/4
0 = x^2 + y^2 - 17y - 289/4

Let's substitute x^2 + y^2 from the first equation into the second equation:

0 = 81 - 17y - 289/4
289/4 - 81 = 17y
(289-324)/4 = 17y
-35/4 = 17y
y = -35/68

Substituting y back into the first equation:
81 = x^2 + (-35/68)^2
81 = x^2 + 1225/4624
81 - 1225/4624 = x^2
(321744 - 1225) / 4624 = x^2
(320519/4624) = x^2
x^2 = 69.296

Taking the square root of both sides to solve for x:
x = ± √69.296

Since we are looking for the point in the first quadrant, we take the positive square root:

x = √69.296

Now we can determine the polar coordinates (r, theta) of point P:

r = √(x^2 + y^2)
r = √(69.296 + (-35/68)^2)
r = √(69.296 + 1225/4624)
r = √(69.296 + 0.264)
r = √(69.56)
r ≈ 8.342

To find theta, we can use the formula:

theta = arctan(y/x)
theta = arctan((-35/68) / (√69.296))
theta ≈ arctan(-0.5102)
theta ≈ -0.4663

Since theta should be between 0 and 2π, we add 2π to the negative theta:

theta ≈ -0.4663 + 2π
theta ≈ 5.816

So the polar coordinates (r, theta) of point P are approximately (8.342, 5.816).

To solve this problem, we need to find the coordinates of the point P where the two circles intersect. We'll use the information given about the circles' centers and radii.

First, let's find the equation of the circle C centered at the origin with radius 9. In polar coordinates, a circle centered at the origin is defined by the equation r = a, where a is the radius. So, the equation of circle C is r = 9.

Next, let's find the equation of the circle K with one end of its diameter at the origin (0, 0) and the other end at the point (0, 17). The midpoint of the diameter will be the center of the circle. So, the center of circle K is the midpoint of the line segment joining the origin and (0, 17), which is ((0+0)/2, (0+17)/2) = (0, 8.5). The radius of circle K is half the length of the diameter, which is 17/2 = 8.5.

Therefore, the equation of circle K is r = 8.5.

Now, let's find the points of intersection between the two circles. We can solve the equations r = 9 and r = 8.5 simultaneously to find the common values of r.

r = 9
r = 8.5

Substituting r = 8.5 into the first equation, we get:

8.5 = 9

Since this is not true, it means the two circles do not intersect.

Therefore, there are no points P that satisfy the given conditions, and we cannot find the values of r and theta.