Answer to Question #234832 in Calculus for Jese Junior

Question #234832

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)? 


1
Expert's answer
2021-09-14T11:26:51-0400

a)

F(x,y)=x22xy+y26Fx(x,y)=2x2yFy(x,y)=2x+2ydydx=Fx(x,y)Fy(x,y)=2x2y2x+2ydydx=2y2x2x2y    dxdy=2x2y2y2xF(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=dydxm=2y2x2x2yAt(2,3)m=23222223m=6446m=22m=1m= \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


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