   Chapter 3.5, Problem 80E

Chapter
Section
Textbook Problem

The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 ≤ 5. If the point (–5. 0) is on the edge of the shadow, how far above the x-axis is the lamp located? To determine

To find: How far above x-axis is the lamp located.

Explanation

Given:

The lamp located three unit from the right of y axis,

The shadow created by elliptical region x2+4y25.

The point (5,0) is on the edge of the shadow.

Calculation:

Let h be the height of the lamp.

Let (a,b) be an point of the tangency through the point (3,h) and (5,0).

The slope of the line passing through the point (3,h) and (5,0),

m=0h53=h8

Thus, the slope of the tangent at (a,b) is m=h8 (1)

Consider the equation of the ellipse x2+4y2=5,

Differentiate with respect to x,

ddx(x2)+4ddx(y2)=ddx(5)2x+4(2ydydx)=0dydx=2x8ydydx=x4y

That is, the slope of the tangent to the ellipse is, dydx=x4y

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