For what positive value of c does the line y = -x + c intersect the circle x^2 + y^2 = 1 in exactly one point?

Guest Nov 26, 2019

#1**+1 **

For what positive value of c does the line y = -x + c intersect the circle x^2 + y^2 = 1 in exactly one point?

Sub -x + c in for y in the circular equation and we have that

x^2 + (-x + c)^2 = 1 simplify

x^2 + c^2 - 2cx + x^2 = 1

2x^2 - 2cx + (c^2 - 1) = 0

If these intersect in exactly one point, the discriminant must = 0....so....

(2c)^2 - 4(2)(c^2 - 1) = 0

4c^2 - 8c^2 + 8 = 0

-4c^2 + 8 = 0 subtract 8 from both sides

-4c^2 = -8 divide both sides by -4

c^2 = 2 take the positive root

c = √2

Here's a graph : https://www.desmos.com/calculator/lithgiri0g

CPhill Nov 26, 2019