Question

P= (x,y) is an arbitrary point on the circle x^2+y^2=36

1. Express the distance d from P to the point A= (8,0) as a function of the x-coordinate of P

2. What are the domain and range of this function d(x)?

Answer 1

Let (x,y) be an arbitrary point on the circle.

Then d = √[x-3)2 + (y-0)2] = √[x2 - 6x + 9 + y2]

Since x2 + y2 = 1, y2 = 1-x2.

So, d = √[x2 - 6x + 9 + 1 - x2]

d = √(-6x+10)

Domain: x is the x-coordinate of a point on the circle centered at (0,0) with radius 1. So, -1≤x≤1.

But, for d to be defined, we need -6x+10 ≥ 0. So, x ≤ 5/3 (True for all x in [-1,1]).

Domain = [-1,1]

Range: -1 ≤ x ≤ 1 So, 6 ≥ -6x ≥ -6

16 ≥ -6x+10 ≥ 4

4 ≥ √(-6x+10) ≥ 2 That is, 2 ≤ d ≤4

Range: [2,4]

Explanation: