Question

Find the values of $m$ for which the line $y=mx-1$ is tangent to the parabola $y=(x-2{)}^2+4$.

Answer 1

Given $y=mx-1$ is tangent to the parabola $y=(x-2{)}^2+4$.

Then the equation,

$mx-1=(x-2{)}^2+4$ will have two equal roots.

or, $mx-1=x^2-4x+8$

or, $x^2-x(4+m)+9=0$.....(1).

Now equation (1) will've two equal roots if,

$(-4-m{)}^2-\mathrm{4.9.1}=0$

or, $(m+4{)}^2=36$

or, $m=-4\pm 6$

or, $m=2,-10$.