   Chapter 3.7, Problem 67E

Chapter
Section
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 fine has x-intercept a 2 / p and y –intercept b 2 / p .(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.

To determine

(a)

To show:

The tangent line has x- intercept a2/pand y-intercept b2/q

Explanation

1) Concept:

Find the equation of the tangent line by using slope-point formula, and from this calculate y- intercept and then at y=0find the x- intercept.

2) Calculation:

Equation of ellipse is x2a2+y2b2=1

Differentiate with respect to x

ddx(x2a2+y2b2)=ddx(1)

Simplify,

2xa2+2yy'b2=0

2yy'b2=-2xa2

y'=-xb2ya2

Which is slope

Therefore,

m=-xb2ya2

By using the slope-point formula, the equation of the tangent line passing through point (p, q) is

y-q=-b2pa2q(x-p)

y=-b2pa2qx+b2p2a2q+q

y=-b2pa2qx+b2p2+a2q2a2q

The last term is called y- intercept

Consider y- intercept

b2

To determine

b)

To show:

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

To determine

c)

To show:

The triangle formed by a tangent line and the coordinate axes has minimum area ab

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