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.
To find the polar coordinates (r, θ) of the point of intersection P in the first quadrant, we need to determine the angle θ and the radius r.
First, let's find the coordinates of the point P.
Since circle C has its center at the origin (0,0) and radius 9, any point on circle C can be represented as (9cos(θ), 9sin(θ)), where θ is the angle between the positive x-axis and the line segment connecting the origin to the point.
Now, let's find the coordinates of the point (0,17) on circle K. Since this point lies on the diameter of circle K with one end at the origin (0,0), we can find its coordinates by dividing the coordinates by 2. So, the coordinates of the point (0,17) are (0/2, 17/2) = (0, 8.5).
To find the angle θ, we need to equate the y-coordinates of the points on the two circles:
9sin(θ) = 8.5
sin(θ) = 8.5/9
θ = arcsin(8.5/9)
Note: Since we want the point P to lie in the first quadrant, the angle θ in the polar coordinate system should be positive. Therefore, we take the positive value of arcsin(8.5/9).
Next, we can find the radius r by using the x-coordinate of the point P on circle C:
x-coordinate of P = 9cos(θ)
Therefore, the polar coordinates (r, θ) of the point P in the first quadrant are:
r = Square root of [(9cos(θ))^2 + (9sin(θ))^2]
θ = arcsin(8.5/9)
By substituting the value of θ into the equation for r, we can find the value of r.
I hope this explanation helps you to understand how to find the polar coordinates of the point P.