Answer
Slope = $\frac{-9}{13}$
Work Step by Step
We use implicit differentiation:
$2(x^2+y^2)^2 = 25(x^2-y^2)$
$4(x^2+y^2) * (2x + 2y \frac{dy}{dx}) = 25 (2x-2y*\frac{dy}{dx})$
$8y(x^2+y^2)* \frac{dy}{dx} + 8x(x^2+y^2) = 50x - 50y * \frac{dy}{dx}$
$(8y(x^2+y^2)+50y)* \frac{dy}{dx} = 50x-8x(x^2+y^2)$
$ \frac{dy}{dx} = \frac{50x-8x(x^2+y^2)}{8y(x^2+y^2)+50y}$
$\frac{dy}{dx} |_{(3,1)} = \frac{50*3-8*3(3^2+1^2)}{8*1(3^2+1^2)+50*1} = \frac{-9}{13}$
