Question

A circle c has center at the origin and radius 5. another circle k has a diameter with one end at the origin and the other end at the point (0,18). 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,θ) be the polar coordinates of p, chosen so that r is positive and 0≤θ≤2. find r and θ.

Answer 1

we know that

circle c
the center is the point (0,0)
radius r=5 units
equation of a circle c is
x²+y²=5²----------> x²+y²=25

circle k
has a diameter with one end at the origin and the other end at the point (0,18)
let
A (0,0) B(0,18)
the distance between A and B is the diameter
diameter=18----------> radius r=18/2-------> r=9 units

the center of circle k is the midpoint A and B
xm=0
ym=(18+0)/2=9
the center is the point (0,9)

the equation of a circle k is
x²+(y-9)²=9²----------> x²+(y-9)²=81

using a graph tool----------> calculate the point of intersection of circle c and circle k which lies in the first quadrant

see the attached figure
the solution is the point p (4.803,1.389)

calculate the polar coordinates of p---------> (r,θ)
r=√[(4.803)²+(1.389)²]--------> r=5 units

tan θ=1.389/4.803-------> tan θ=0.28919
θ=arctan (0.28919)--------> θ=16.13°----------> 0.09pi

the solution is
r=5 units
θ=16.13° (0.09pi)