A circle C has center at the origin and radius 4. Another circle K has a diamete

Question

A circle C has center at the origin and radius 4. Another circle K has a diameter with one end at the origin and the other end at the point (0, 14). 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, ?) be the polar coordinates of P, chosen so that r is positive and 0 ? ? ? 2. Find r and ?. A circle C has center at the origin and radius 4. Another circle K has a diameter with one end at the origin and the other end at the point (0, 14). 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, ?) be the polar coordinates of P, chosen so that r is positive and 0 ? ? ? 2. Find r and ?. A circle C has center at the origin and radius 4. Another circle K has a diameter with one end at the origin and the other end at the point (0, 14). 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, ?) be the polar coordinates of P, chosen so that r is positive and 0 ? ? ? 2. Find r and ?.

Explanation / Answer

Circle C : centered at 0,0 and radius = 4
So, x^2 + y^2 = 16

Circle K :
diameter ends are 0,0 and 0,14
So, center is at midpoint i.e 0,7
And radius = 7

Using (x-h)^2 + (y-k)^2 = r^2

x^2 + (y - 7)^2 = 7^2

x^2 + y^2 = 16
x^2 + (y - 7)^2 = 49

Subtracting :
y^2 - (y - 7)^2 = 16 - 49

y^2 - (y^2 - 14y + 49) = -33

14y - 49 = -33

y = 8/7

And Using x^2 + y^2= 16,
x^2 + 64/49 = 16
x^2 = 720/49
x = 12sqrt(5)/7

So, we have
(12sqrt5/7 , 8/7)

Now, we have to polar form this point....
Clearly r = sqrt(x^2 + y^2), i.e sqrt(16) = 4 ----> ANS

And theta = arctan(y/x)

= arctan(8/(12sqrt(5))

= 0.28975 radians ----> ANS

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