What is the slope of b^2 x^2 + a^2 y^2 = b^2 a^2 at (α,β) ?

Question

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Answer 1

Start by taking derivative of both sides on a term by term basis, getting

2a^2(y)dy/dx + 2b^2(x) = 0, the right side zero because the derivative on a constant is 0

Divide both sides through by 2 and rearranging to get dy/dx = - (b/a)^2(x/y)

Plug in σ and β to get answer of - (b/a)^2(α/β)

Answer 2

Hi Friday,

The slope of b²*x²+a²*y² = b²*a² at (α,β) is (-α*b²)/(β*a²). To determine this, we have to find what dy/dx (another way of saying change in y divided by change in x, or the slope) of the given equation. To do this we have to differentiate the expression in terms of x.

1. Differentiate the equation in terms of x:
2. d/dx of (b²*x²+a²*y²) = d/dx of (b²*a²)
3. d/dx of (b²*x²) + d/dx of (a²*y²) = d/dx of (b²*a²)
4. Now we apply the rule of differentiation to determine that:
5. 2x*b² + 2y*(dy/dx)*a² = 0 <----- derivative of a constant is 0
6. Now we have to isolate dy/dx
7. 2y*(dy/dx)*a² = -2x*b²
8. dy/dx = (-2x*b²)/(2y*a²)
9. dy/dx = (-x*b²)/(y*a²)
10. To find the slope at (α,β) we just plug that into the equation of dy/dx
11. dy/dx = (-α*b²)/(β*a²)

Thus, the slope of b²*x²+a²*y² = b²*a² at (α,β) is (-α*b²)/(β*a²).

If you have anymore questions, just ask me!

I hope this helps,

Manu