Acircle c has center at the origin and radius 8. 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)(r,θ) be the polar coordinates of p, chosen so that rr is positive and 0≤θ≤2.0≤θ≤2. find rr and θθ.

In order to find the point of intersection, we need to find the equations of the two circles.

Once we know the center and radius of a circle, we can write its expression as

We know that circle c has center and radius 8, so we immediately deduce its equation:

As for circle k, we know that the segment with endpoints the origin and is a diameter. So, the center is the midpoint of this segment, i.e. . Moreover, the segment with endpoints the origin and the center is a radius, which is 9 units long.

So, the equation for circle k is

So, the intersection points are given by the system

If we subtract the first equation from the second, we have

Which expands to

Which yields the following values for x:

We are interested in the point belonging to the first quadrant, so we only accept the positive solution. So, we have

To convert this point in polar coordinates, use the definitions

We already know that , because the point lies on circle c.

And finally, we have

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step-by-step explanation: