The equation for C is \(x^{2} + y^{2} = r^{2}\) and P is a point (a, 0)

on the x-axis with a \(\neq \pm r.\)

The point on the circle C is(x, y).

Obtain the distance between (x, y) and (a, 0) as follows:

\(d =\sqrt{(x - a)^2 + (y - 0)^2}\)

\(=\sqrt{(x-a)^{2}+y^{2}}\)

Now, obtain the partial derivatives as follows,

\(d_{x} = 0\)

\(\frac{2(x-a)}{2\sqrt{(x-a)^{2}+y^{2}}}=0\)

\(x = a\)

\(d_{y} = 0\)

\(\frac{2y}{2\sqrt{(x-a)^{2}+y^{2}}}=0\)

\(y = 0\)

Thus, the critical point is(a, 0). However, this point doesn’t lie on the circle C.

We know the endpoints on the circles are (-r, 0) and (r, 0).

Thus, the distance becomes

At (-r, 0),

\(d = \sqrt{(-r-a)^{2}+(0)^{2}}\)

\(=\sqrt{(r+a)^{2}}\)

\(= r + a\)

and at (r, 0),

\(d=\sqrt{(-r-a)^{2}+(0)^{2}}\)

\(=\sqrt{(r-a)^{2}}\)

\(= r - a\)

Apart from these two points, no other point gives these distances but between \(r + a, r - a.\)

Also, these two points lie on the line from (a, 0) that passes through the center of C.

Thus, the maximum distance from the point P (a, 0) to the point \(X (x, y) on C is r + a\)

and the minimum distance from the point P (a, 0) to the point \(X (x, y) on C is r - a.\)

Hence, the maximum and minimum distances from P toa point X on C occur when the line XP goes through the center of C.

on the x-axis with a \(\neq \pm r.\)

The point on the circle C is(x, y).

Obtain the distance between (x, y) and (a, 0) as follows:

\(d =\sqrt{(x - a)^2 + (y - 0)^2}\)

\(=\sqrt{(x-a)^{2}+y^{2}}\)

Now, obtain the partial derivatives as follows,

\(d_{x} = 0\)

\(\frac{2(x-a)}{2\sqrt{(x-a)^{2}+y^{2}}}=0\)

\(x = a\)

\(d_{y} = 0\)

\(\frac{2y}{2\sqrt{(x-a)^{2}+y^{2}}}=0\)

\(y = 0\)

Thus, the critical point is(a, 0). However, this point doesn’t lie on the circle C.

We know the endpoints on the circles are (-r, 0) and (r, 0).

Thus, the distance becomes

At (-r, 0),

\(d = \sqrt{(-r-a)^{2}+(0)^{2}}\)

\(=\sqrt{(r+a)^{2}}\)

\(= r + a\)

and at (r, 0),

\(d=\sqrt{(-r-a)^{2}+(0)^{2}}\)

\(=\sqrt{(r-a)^{2}}\)

\(= r - a\)

Apart from these two points, no other point gives these distances but between \(r + a, r - a.\)

Also, these two points lie on the line from (a, 0) that passes through the center of C.

Thus, the maximum distance from the point P (a, 0) to the point \(X (x, y) on C is r + a\)

and the minimum distance from the point P (a, 0) to the point \(X (x, y) on C is r - a.\)

Hence, the maximum and minimum distances from P toa point X on C occur when the line XP goes through the center of C.