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# (See the Guided Projects for additional applications of ISBN: 9780321570567 2

## Solution for problem 83E Chapter 10.4

Calculus: Early Transcendentals | 1st Edition

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Problem 83E

(See the Guided Projects for additional applications of conic sections.)Reflection property of parabolas Consider the parabola y = x2/(4p) with its focus at F(0, p) (see figure). We know that light is reflected from a surface in such a way that the angle of incidence equals the angle of reflection. The goal is to show that the angle between the ray ?and the tangent line L (? in the figure) equals the angle between the line PF and L (? in the figure). If these two angles are equal, then the reflection properly is proved because ? is reflected through F.a. Let P(x0, y0) be a point on the parabola. Show that the slope of the line tangent to the curve at P is tan ? = x0/(2p).b. Show that tan ?= (p ? y0)/x0.c. Show that ? = ?/2 ? ?; therefore, tan ? = cot ?.d. Note that ? = ? + ?. Use the tangent addition formula to show that tan ? = tan ? = 2p/x0.e. Conclude that because ? and ? are acute angles, ? = ?.

Step-by-Step Solution:

Solution 83EStep 1 of 5:a) In this problem we need to Show that the slope of the line tangent to the curve at P is tan(.Given parabola is y = with its focus at F (0 , p).Let p() be a point on the parabola.Let be an inclination of the tangent line , then the slope of the tangent line is tan(We know that ,slope is denoted by = m = tan(Consider , y = then differentiate both sides with respect to x we get , = () = , since cwhere c is constant. = .Therefore , the slope of the tangent line at the point p() is : = m = tan(= Step 2 of 5:b) In this problem we have to show that tan = .The given figure is ; Clearly , from the figure : tan() = -tan() = = = , since OF = , OB||OP = p and OP = .Therefore , tan() = .Step 3 of 5:c) In this we need to show that = /2 ; therefore, tan = cot .From the figure , it is clear that the line ‘l’ makes an angle with the horizontal line , and the tangent line ‘L’ makes angle with the horizontal line , and also the angle between the line ‘l’ and the tangent line ‘L’ is Therefore , - . - . = .Therefore , = .

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##### ISBN: 9780321570567

The full step-by-step solution to problem: 83E from chapter: 10.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. The answer to “(See the Guided Projects for additional applications of conic sections.)Reflection property of parabolas Consider the parabola y = x2/(4p) with its focus at F(0, p) (see figure). We know that light is reflected from a surface in such a way that the angle of incidence equals the angle of reflection. The goal is to show that the angle between the ray ?and the tangent line L (? in the figure) equals the angle between the line PF and L (? in the figure). If these two angles are equal, then the reflection properly is proved because ? is reflected through F.a. Let P(x0, y0) be a point on the parabola. Show that the slope of the line tangent to the curve at P is tan ? = x0/(2p).b. Show that tan ?= (p ? y0)/x0.c. Show that ? = ?/2 ? ?; therefore, tan ? = cot ?.d. Note that ? = ? + ?. Use the tangent addition formula to show that tan ? = tan ? = 2p/x0.e. Conclude that because ? and ? are acute angles, ? = ?.” is broken down into a number of easy to follow steps, and 182 words. Since the solution to 83E from 10.4 chapter was answered, more than 247 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: show, tan, angle, Tangent, figure. This expansive textbook survival guide covers 85 chapters, and 5218 solutions.

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