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

kuCAu 2020-12-14 Answered
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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Expert Answer

Nichole Watt
Answered 2020-12-15 Author has 100 answers

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=(xa)2+(y0)2
=(xa)2+y2
Now, obtain the partial derivatives as follows,
dx=0
2(xa)2(xa)2+y2=0
x=a
dy=0
2y2(xa)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=(ra)2+(0)2
=(r+a)2
=r+a
and at (r, 0),
d=(ra)2+(0)2
=(ra)2
=ra
Apart from these two points, no other point gives these distances but between r+a,ra.
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 ra.
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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