   Chapter 3.5, Problem 74E

Chapter
Section
Textbook Problem

(a) Where does the normal line to the ellipse x2 – xy + y2 = 3 at the point (–1, 1) intersect the ellipse a second time?(b) Illustrate part (a) by graphing the ellipse and the normal line.

To determine

To find: The point such that the normal line intersect the ellipse at second time and to sketch the equation of the ellipse and the normal line.

Explanation

Given:

The equation of the ellipse is x2xy+y2=3 and the point is (1,1).

Derivative rules:

Product rule: ddx(fg)=fddx(g)+gddx(f)

Chain rule: dydx=dydududx

Calculation:

Obtain the slope of the normal line to the ellipse.

Differentiate x2xy+y2=3 implicitly with respect to x,

ddx(x2xy+y2)=ddx(3)ddx(x2)ddx(xy)+ddx(y2)=ddx(3)2xddx(xy)+ddx(y2)=0

Apply the product rule and the chain rule,

2xddx(xy)+ddx(y2)=02x[xddx(y)+yddx(x)]+ddy(y2)dydx=02x[xdydx+y]+2ydydx=02xyxdydx+2ydydx=0

Combine the dydx on one side of the equation,

(2yx)dydx=y2xdydx=y2x2yx

Thus, the derivative of the equation is dydx=y2x2yx

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