   Chapter 4.7, Problem 67E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Consider the tangent line to the ellipse x 2 a 2 + y 2 b 2 = 1 at a point (p, q) in the first quadrant.(a) Show that the tangent line has x-intercept a2/p and y-intercept b2/q.(b) Show that the portion of the tangent line cut off by the coordinate axes has minimum length a + b.(c) Show that the triangle formed by the tangent line and the coordinate axes has minimum area ab.

(a)

To determine

To show: The tangent line x intercept a2p and y intercept b2q

Explanation

Given:

The equation of the ellipse is x2a2+y2b2=1

The tangent line at a point (p,q) in 1st quadrant of the ellipse.

Proof:

The equation of the ellipse is x2a2+y2b2=1.

Consider the ellipse as shown in the figure,

Differentiate with respect to x

2xa2+2yb2dydx=0dydx=b2xa2y

at(p,q) dydx=b2pa2q

The equation of the tangent line is

yq=b2pa2q(xp)y=b2pa2qx+b2p2a2q+qy=b2pa2qx+b2p2+a2q2a2q

Substitute x=0 in y=b2pa2qx+b2p2+a2q2a2q

y=

(b)

To determine

To show: The portion of the tangent line cut off by the coordinate axes minimum length a+b

(c)

To determine

To show: The triangle formed by the tangent line and the coordinate axes has minimum area ab

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Study Guide for Stewart's Multivariable Calculus, 8th 