   Chapter 2.6, Problem 58E

Chapter
Section
Textbook Problem

# (a) Where does the normal line to the ellipse x 2 − x y   +   y 2 =   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

(a)

To Find:

Coordinate of normal line when intersection the ellipse second time.

Explanation

1) Concept:

Implicit differentiation and Point slope form of equation of line are the concepts used.

2) Formula:

i. Sum or difference rule

ddxfx ±gx=ddx fx± ddxgx ;

ii. Derivative of constant

ddxc=0,

iii. Product rule

ddxfx*gx=fxddxgx+gxddx fx

iv. Power rule

ddxxn=n xn-1

3) Given:

x2-xy+y2=3

4) Calculations:

To find the next coordinate points of normal line to ellipse, find the derivative of given equation of ellipse at point (-1,1).

Differentiating given expression implicitly with respect to x,

ddxx2-xy+y2=ddx3

2x-xdydx+y1+2ydydx=0

2x-xdydx-y+2ydydx=0

-xdydx+2ydydx+2x-y=0

-x+2ydydx

To determine

(b)

To illustrate:

The graph of normal line and ellipse.

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