How to calculate tangent of function, which has to cross certain value on x axis

Question

For example, let's take a simple function $f(x) = x^2-2$.

How to get a tangent, which has to cross given value at x axis, for example $x = 2$, like that:

Thank you very much for any help.

Answer 1

We come up with a general function $T_{a}$ which is the tangent line to $f$ at the point $x=a$. Indeed, $$ T_{a}(x)=f(a)+f'(a)(x-a)=(a^{2}-2)+2a(x-a)\tag{1} $$ We want $T(2)=0$. Solve for $a$ using $(1)$ and the resulting $T_a$ will be your required tangent line.