Finding coordinates at which tangent lines intersect

Question

Tangent lines are drawn to the function $f(x)=x^2-4/x$ at the points $(-1,5)$ and $(1,-3)$. Find the coordinates of the point at which the tangent lines intersect.

I'm not sure how to approach this question dealing with derivatives. Can anyone help?

Answer 1

The equation of the tangent line at a point $(x_0,f(x_0))$ of a curve defined by $y=f(x)$ where $f$ is derivable at $x_0$, is $y=f(x_0)+f'(x_0)\cdot (x-x_0)$

Let's now apply that to our problem the first tangent line is $y=5+2(x+1)$ and the second is $y=-3+6(x-1)$ and their intersection $(x_A,y_A)$ is such that $5+2(x_A+1)=-3+6(x_A-1)$ whose solution is $x_A=4$. This gives $y_A=15$