Question

A curve has an equation given by x2 – 2xy + y2 = 6.

a) Obtain an expression for dx dy in terms of x and y.

b) What is the gradient of the curve at (2, 3)? 

Answer 1

a)

$F(x,y)=x^2 – 2xy + y^2 -6\\ F'_x(x,y)= 2x-2y\\ F'_y(x,y)= -2x+2y\\ \frac{dy}{dx}=−\frac{F'_x(x,y)}{F'_y(x,y)}= -\frac{2x-2y}{-2x+2y}\\ \frac{dy}{dx}=\frac{2y-2x}{2x-2y}\\ \implies \frac{dx}{dy}=\frac{2x-2y}{2y-2x}\\$

b)

$m= \frac{dy}{dx}\\ m=\frac{2y-2x}{2x-2y}\\ At (2, 3)\\ m=\frac{2*3-2*2}{2*2-2*3}\\ m=\frac{6-4}{4-6}\\ m=\frac{2}{-2}\\ m=-1$