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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

(a) Use the identity for tan(xy) (see Equation 14b in Appendix D) to show that if two lines L1, and L2 intersect at an angle α, then

tan α = m 2 m 1 1 + m 1 m 2

where m1, and m2 are the slopes of L1 and L2 respectively.

(b) The angle between the curves C1 and C2 at a point of intersection P is defined to be the angle between the tangent lines to C1, and C2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair or curves at each point of intersection.

(i) y = x2 and y = (x – 2)2

(ii) x2y2 = 3 and x2 – 4x + y2 + 3 = 0

(a)

To determine

To show: If two lines intersect at an angle α then tanα=m2m11+m1m2.

Explanation

Given:

Two lines are L1 and L2.

The slope of the lines L1 and L2 are m1 and m2 respectively.

Calculation:

Let ϕ1 and ϕ2 be an inclination angle of the line L1 and L2 respectively.

Here, the slopes of the line L1 and L2 are tanϕ1 and tanϕ2.

The angle between two lines are α=|ϕ2ϕ1|

(b)

To determine

To find: The angle between each pair of curves at each points.

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