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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Given an ellipse x2/a2 + y2/b2 = 1, where a ≠ b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.

(a)

To determine

To find: The equation of the set of all points from which there are two tangents to the curve whose slope are reciprocals.

Explanation

Given:

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

Calculation:

Assume that y=mx+c is a tangent line to the ellipse.

That is, the line intersect exactly one point on the ellipse,

Substitute y=mx+c in x2a2+y2b2=1,

x2a2+(mx+c)2b2=1x2a2+m2x2+c2+2mxcb2=1b2x2+m2x2a2+a2c2+2mxca2=a2b2x2(b2+m2a2)+2mca2x+a2c2a2b2=0

Since the line intersect exactly one point on the ellipse, the discriminant of the equation is zero (b24ac=0).

(2mca2)24(b2+m2a2)(a2c2a2b2)=04m2c2a44(b2a2c2a2b4+m2a4c2m2a4b2)=04m2c2a44b2a2c2+4a2b44m2a4c2+4m2a4b2=04b2a2c2+4a2b4+4m2a4b2=0

Simplify the terms,

4a2b2(c2+b2+a2m2)=0

a2m2+b2c2=0

If the point (α,β) lies on the line y=mx+c, then β=mα+c

(b)

To determine

To find: The equation of the set of all points from which there are two tangents to the curve whose slope are negative reciprocals.

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