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 ≠±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.]

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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 ≠±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.]

Nichole Watt · Accepted Answer

The equation for C is x2+y2=r2 and P is a point (a, 0) on the x-axis with a ≠±r. The point on the circle C is(x, y). Obtain the distance between (x, y) and (a, 0) as follows: d=(x−a)2+(y−0)2 =(x−a)2+y2 Now, obtain the partial derivatives as follows, dx=0 2(x−a)2(x−a)2+y2=0 x=a dy=0 2y2(x−a)2+y2=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=(−r−a)2+(0)2 =(r+a)2 =r+a and at (r, 0), d=(−r−a)2+(0)2 =(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.

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

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