Setting up an equation of a line given a coordinate and a graph that the line touches

Question

Say you are given the graph $f(x)=x^2$

You are also given a point $p$, for this example take $p=(-1, -1)$

How do i set up an equation of a line that passes trough $p$, and touches the graph $f(x)=x^2$ somewhere?

Thanks

Answer 1

The equation of a line of gradient $m$ through the point $(-1,-1)$ is $y+1=m(x+1)$. Substitute this into $y=x^2$ to get a quadratic in $x$, if the discriminant of this quadratic is zero then the line will touch the curve.

$x^2-mx+(1-m)=0$

so

$\Delta=m^2-4(1-m)=0$

so

$m=-2 \pm 2 \sqrt{2}$.