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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

How many lines are tangent to both of the circles x2 + y2 = 4 and x2 + (y – 3)2 = 1? At what points do these tangent lines touch the circles?

To determine

To find: The number of tangent lines to both of the given circles and obtain the points.

Explanation

Given:

The circles x2+y2=4 and x2+(y3)2=1.

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

Where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1,y=y1

Calculation:

Let the equation of tangent be y=mx+c (2)

Substitute y=mx+c in x2+y2=4,

x2+(mx+c)2=4x2+m2x2+c2+2mcx=4x2(1+m2)+2mcx+c24=0

The line touches the circle discriminant will be zero (D=0).

The quadric equation is b24ac=0 from equation x2(1+m2)+2mcx+c24=0.

(2mc)24(1+m2)(c24)=04m2c24(c24+m2c24m2)=04(m2c2c2+4m2c2+4m2)=0

4m2c2+4=0 (2)

Substitute y=mx+c in x2+(y3)2=1.

x2+(mx+c3)2=1x2+m2x2+(c3)2+2m(c3)x=1x2(1+m2)+2m(c3)x+c2+96c=1x2(1+m2)+2m(c3)x+c2+86c=0

The line touches the circle discriminant will be zero (D=0).

The quadric equation is b24ac=0 from equation,

x2(1+m2)+2m(c3)x+c2+86c=0

(2m(c3))24(1+m2)(c26c+8)=04m2(c3)24(c26c+8+m2c26m2c+8m2)=04m2(c2+96c)4(c26c+8+m2c26m2c+8m2)=0m2c2+9m26m2cc2+6c8m2c2+6

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