kuCAu

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$\ne \pm r,$ 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.]

on the x-axis with a

Nichole Watt

Skilled2020-12-15Added 100 answers

The equation for C is

on the x-axis with a

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

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

Now, obtain the partial derivatives as follows,

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),

and at (r, 0),

Apart from these two points, no other point gives these distances but between

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

and the minimum distance from the point P (a, 0) to the point

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

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