Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by x2 + y2 = r2 and P is a point (a,0) on the x-axis with a neq pm r, use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]

Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by x2 + y2 = r2 and P is a point (a,0) on the x-axis with a neq pm r, use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]

Question
Alternate coordinate systems
asked 2020-12-14
Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by
\(x2 + y2 = r2\) and P is a point (a,0)
on the x-axis with a \(\neq \pm r,\) use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]

Answers (1)

2020-12-15
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.
0

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