Question

A tangent is a line that touches a circle at exactly one point. For what values of k will the line y= x+k be tangent to the circle x^2+ y^2 = 25?

I know I'm suppose to use the properties of a discrimminate. But how

Answer 1

the intersection(s) will be where

x^2 + (x+k)^2 - 25 has one solution. That is, where the discriminant is zero.

x^2 + x^2 + 2kx + k^2-25 = 0
2x^2 + 2kx + (k^2-25) = 0
The discriminant is

(2k)^2 - 4(2)(k^2-25)
4k^2 - 8k^2 + 200 = 0
k^2 = 50
k = ±√50

So, check out the graphs at

http://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3D25%2C+y%3Dx%2B%E2%88%9A50%2C+y%3Dx-%E2%88%9A50