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)?
a)
F(x,y)=x2–2xy+y2−6Fx′(x,y)=2x−2yFy′(x,y)=−2x+2ydydx=−Fx′(x,y)Fy′(x,y)=−2x−2y−2x+2ydydx=2y−2x2x−2y ⟹ dxdy=2x−2y2y−2xF(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}\\F(x,y)=x2–2xy+y2−6Fx′(x,y)=2x−2yFy′(x,y)=−2x+2ydxdy=−Fy′(x,y)Fx′(x,y)=−−2x+2y2x−2ydxdy=2x−2y2y−2x⟹dydx=2y−2x2x−2y
b)
m=dydxm=2y−2x2x−2yAt(2,3)m=2∗3−2∗22∗2−2∗3m=6−44−6m=2−2m=−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=-1m=dxdym=2x−2y2y−2xAt(2,3)m=2∗2−2∗32∗3−2∗2m=4−66−4m=−22m=−1
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