Finding the condition that m and c satisfy with a given circle

Question

How can I find the condition that m and c satisfy if y = mx + c touches the circle x² + y² - 2ax = 0

Answer 1

I'm assuming that "touches" means tangent to the circle.

Your line has the equation y = mx + c

Your circle has the equation x² + y² - 2ax = 0

If we substitute the line's equation into the circle's equation, we get:

x² + (mx + c)² - 2ax = 0

(1+m²)x² + 2(cm-a)x + c² = 0

We can use the quadratic formula to solve this equation. Since the line is tangent, the quadratic formula must have one solution and thus the discriminant is 0. So the quadratic formula simplifies to:

x = -b/2a

(Note that this is the quadratic formula. The a here is the leading coefficient and not the a of the formulas above.)

x = -2(cm-a) / 2(1+m²) = (a-cm) / (1+m²)

We put this solution back into our circle's equation. After simplification, it becomes:

m = (a²-c²) / 2ac

Answer 2

circle y²=-x²+2ax

line y²=m²x²+2mcx+c²

the equations are set equal to find intersection

have quadratic in one variable

(m²+1)x²+2(mc-a)x+c²=0

setting the discriminant to zero will make the solution

a one point solution meaning line tangent to circle

need 4m²c²-8mac+4a²-4m²c²-4c²=0

same as

m=(a²-c²)/2ac

choosing a=2 and c=1,,,m=3/4

graphing shows line tangent to circle.