Question

If the line x+y=r is tangent to the circle x²+y²=r, where r is positive, then r = ?

Please explain, I know the answer.
Thanks!

Answer 1

r = 2.

Explanation:

We can find the slope of the tangent using implicit differentiation:

x^2 + y^2 = r

2x + dy/dx * 2y = 0

dy/dx = -2x / 2y = -x/y = the slope of the tangent.

We have the line x + y = r

y = -x + r

This has a slope of -1 , so:

-x/y = -1

-x = -y

x = y

Substituting in x^2 + y^2 = r and x + y = r:-

2x^2 = r

2x = r

2x^2 - 2x = 0

2x(x - 1) = 0

x = 1

So r = 2x = 2.