Chapter 2.P, Problem 11P

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

How many lines are tangent to both of the circles

To determine

**To find: **

how many lines are tangent to both thecircles

Explanation

Graph of both circles and tangent lines are shown in the following figure:

**1) Given:**

**2) Calculations:**

Let C_{1} be the _{2} be the

From above figure, the line y = 2 is tangent to both the circles C_{1} and C_{2} at the point (0, 2).

Let us use implicit differentiation to find the slope of the tangent line.

Differentiate equation of first circle with respect to x:

Divide both sides by 2,

Subtract x from both sides,

Divide both sides by y,

Thus the slope of tangent to C_{1} at point (x, y) is

Similarly, differentiate equation of second circle with respect to x.

Divide both sides by 2,

Subtract x from both sides,

Divide both sides by

Thus the slope of tangent to C_{2} at point (x, y) is

Thus, from (2) and (4), we get slope of tangent line m = 0 for both circles at point (0, 2).

Now substitute m = 0 and point (

Therefore, the equation of the tangent line is y = 2.

From the figure, tangent line L_{1} passing through the points (a, b) & (c, d**#x0029; and if such a line exist then itâ€™s reflection about y axis is another tangent line L_{2}.

Also, the slope of L_{1} is same at point (a, b) and (c, d).

To find slope of circle C_{1} at point (a, b), substitute x = a and y = b in

Similarly, to find slope of circle C_{2} at point (c, d), substitute x = c and y = d in

Substitute x = c and y = d in

Since (c, d) is a point on C_{2} then it becomes,

Substitute

Factor out

Now substitute

By taking square root of both sides,

Substitute

Now find y intercept

Use slope intercept form:

Substitute

Similarly, find y intercept

Use slope intercept form:

Substitute

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