Find the Equation of the Tangent Line to the Curve (A lemniscate) $2(x^2+y^2)^2=25(x^2-y^2)$ at the point $(3,-1)$. The equation of this tangent line can be written in the form $y=mx+b$ where:
$m=$________
$b=$________
My work is in the file attached. I assumed I should factor out the equation first, but maybe that's where I went wrong. 
Best Answer
Applying Implicit Differentiation, $$2\cdot2(x^2+y^2)\cdot\left(2x+2y\frac{dy}{dx}\right)=25\cdot\left(2x-2y\frac{dy}{dx}\right)$$
So, at $(3,-1)$ it reduces to $$80\left(3-\frac{dy}{dx}\right)=50\left(3+\frac{dy}{dx}\right)$$