Final answer:
To find the slope of the tangent line to the curve, differentiate the equation with respect to x, substitute the given point, and solve for dx/dx.
Step-by-step explanation:
To find the slope of the tangent line to the curve, we need to find the derivative of the curve and evaluate it at the given point. Let's start by differentiating the equation with respect to x.
First, we need to expand and simplify the expression: 2(x^2 + y^2)^2 = 25(x^2 - y^2). This gives us 2x^4 + 4x^2y^2 + 2y^4 = 25x^2 - 25y^2.
Now, differentiate both sides of the equation with respect to x:
8x^3 + 8xy^2(dx/dx) + 4y^2x + 4y^2(dx/dx) = 50x - 50y^2(dx/dx)
Simplify this equation by substituting the given point coordinates: (3, -1). Solve for dx/dx, which represents the slope of the tangent line at that point.