Algebra 2
Algebra 2
1st Edition
ISBN: 9780078884825
Author: McGraw-Hill/Glencoe
Publisher: Glencoe/McGraw-Hill School Pub Co
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Chapter 10, Problem 51SGR
To determine

To find: The where should light hit the mirror so that light will be reflected to (0,−5).

Expert Solution & Answer
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Answer to Problem 51SGR

The light sourceshould hit the mirror at point (40−2455,45−1255) so that it will be reflected to (0,−5).

Explanation of Solution

Given information:

  y29−x216=1

Light source (−10,0)

Light will be reflected to (0,−5).

Calculation:

Consider a hyperbolic mirror which is modeled by the upper branch of the hyperbola

  y29−x216=1                  .........(1)

The mirror reflects light rays directed at one focus towards the other focus.

Compare equation (1) y29−x216=1 with standard form of a vertical hyperbola with center at origin y2a2−x2b2=1 as shown below:

  a2=9a=3

  b2=16b=4

Use relation between a, b and c and compute the value of c as follows,

  c2=a2+b2c2=9+16c=5

Hence, foci of the given hyperbola will be

  (h,k±c) = (0,0±5)= (0,−5)= (0,5)

A light source located at (−10,0) should be directed to its one focus (0,5) . So that it will be reflected to other focus (0,−5) . Hence, the path of light source should be the line joining points (−10,0) and (0,5).

Compute the equation of the path of the light source using equation of line passing through two points (x1,y1) and (x2,y2).

  (y−y1)=(y2−y1)(x2−x1)(x−x1)(y−0)=(5−0)(0−(−10))(x−(−10))y=510(x+10)

  y=12(x+10)                  ........(2)

Compute the intersection point of hyperbola (1) y29−x216=1 and line (2) y=12(x+10)  

Substitute y=12(x+10)   in equation (1) and simplify,

  [12(x+10) ]2 9−x216=116[14(x2+20x+100) ]−9x2=1444x2+80x+400−9x2=144

  −5x2+80x+256=0

Use quadratic formula,

  x=−80±802−4×(−5)×2562×(−5)x=40±2455

For x=40+2455 the value of y will be,

  y=12[40+2455+10]y=45+1255

For x=40−2455 the value of y will be,

  y=12[40−2455+10]y=45−1255

So, intersection points of given hyperbola and the path of light source are (40+2455,45+1255) and (40−2455,45−1255)

The point (40+2455,45+1255) will come after the focus (0,5) . Therefore, the light sourceshould hit the mirror at point (40−2455,45−1255) so that it will be reflected to (0,−5).

Therefore, the light sourceshould hit the mirror at point (40−2455,45−1255) so that it will be reflected to (0,−5).

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Chapter 10, Problem 50SGR
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Chapter 10, Problem 52SGR

Chapter 10 Solutions

Algebra 2

Ch. 10.1 - Prob. 1ACYPCh. 10.1 - Prob. 1BCYPCh. 10.1 - Prob. 2CYPCh. 10.1 - Prob. 3CYPCh. 10.1 - Prob. 1CYUCh. 10.1 - Prob. 2CYUCh. 10.1 - Prob. 3CYUCh. 10.1 - Prob. 4CYUCh. 10.1 - Prob. 5CYUCh. 10.1 - Prob. 6CYU
Ch. 10.1 - Prob. 7CYUCh. 10.1 - Prob. 8CYUCh. 10.1 - Prob. 9CYUCh. 10.1 - Prob. 10PPSCh. 10.1 - Prob. 11PPSCh. 10.1 - Prob. 12PPSCh. 10.1 - Prob. 13PPSCh. 10.1 - Prob. 14PPSCh. 10.1 - Prob. 15PPSCh. 10.1 - Prob. 16PPSCh. 10.1 - Prob. 17PPSCh. 10.1 - Prob. 18PPSCh. 10.1 - Prob. 19PPSCh. 10.1 - Prob. 20PPSCh. 10.1 - Prob. 21PPSCh. 10.1 - Prob. 22PPSCh. 10.1 - Prob. 23PPSCh. 10.1 - Prob. 24PPSCh. 10.1 - Prob. 25PPSCh. 10.1 - Prob. 26PPSCh. 10.1 - Prob. 27PPSCh. 10.1 - Prob. 28PPSCh. 10.1 - Prob. 29PPSCh. 10.1 - Prob. 30PPSCh. 10.1 - Prob. 31PPSCh. 10.1 - Prob. 32PPSCh. 10.1 - Prob. 33PPSCh. 10.1 - Prob. 34PPSCh. 10.1 - Prob. 35PPSCh. 10.1 - Prob. 36PPSCh. 10.1 - Prob. 37PPSCh. 10.1 - Prob. 38PPSCh. 10.1 - Prob. 39PPSCh. 10.1 - Prob. 40PPSCh. 10.1 - Prob. 41PPSCh. 10.1 - Prob. 42HPCh. 10.1 - Prob. 43HPCh. 10.1 - Prob. 44HPCh. 10.1 - Prob. 45HPCh. 10.1 - Prob. 46HPCh. 10.1 - Prob. 47STPCh. 10.1 - Prob. 48STPCh. 10.1 - Prob. 49STPCh. 10.1 - Prob. 50STPCh. 10.1 - Prob. 51SRCh. 10.1 - Prob. 52SRCh. 10.1 - 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Prob. 54SRCh. 10.4 - Prob. 55SRCh. 10.4 - Prob. 56SRCh. 10.4 - Prob. 57SRCh. 10.4 - Prob. 58SRCh. 10.4 - Prob. 59SRCh. 10.4 - Prob. 60SRCh. 10.4 - Prob. 61SRCh. 10.4 - Prob. 62SRCh. 10.4 - Prob. 63SRCh. 10.4 - Prob. 64SRCh. 10.4 - Prob. 65SCh. 10.4 - Prob. 66SCh. 10.4 - Prob. 67SCh. 10.4 - Prob. 1MCQCh. 10.4 - Prob. 2MCQCh. 10.4 - Prob. 3MCQCh. 10.4 - Prob. 4MCQCh. 10.4 - Prob. 5MCQCh. 10.4 - Prob. 6MCQCh. 10.4 - Prob. 7MCQCh. 10.4 - Prob. 8MCQCh. 10.4 - Prob. 9MCQCh. 10.4 - Prob. 10MCQCh. 10.4 - Prob. 11MCQCh. 10.4 - Prob. 12MCQCh. 10.4 - Prob. 13MCQCh. 10.4 - Prob. 14MCQCh. 10.4 - Prob. 15MCQCh. 10.4 - Prob. 16MCQCh. 10.4 - Prob. 17MCQCh. 10.4 - Prob. 18MCQCh. 10.4 - Prob. 19MCQCh. 10.4 - Prob. 20MCQCh. 10.5 - Prob. 1CYPCh. 10.5 - Prob. 2CYPCh. 10.5 - Prob. 3CYPCh. 10.5 - Prob. 4CYPCh. 10.5 - Prob. 1CYUCh. 10.5 - Prob. 2CYUCh. 10.5 - Prob. 3CYUCh. 10.5 - Prob. 4CYUCh. 10.5 - Prob. 5CYUCh. 10.5 - Prob. 6CYUCh. 10.5 - Prob. 7CYUCh. 10.5 - Prob. 8CYUCh. 10.5 - Prob. 9CYUCh. 10.5 - Prob. 10PPSCh. 10.5 - 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Linear Algebra and Its Applications (5th Edition)

In the following exercises, solve by completing the square. 73. c212c=13

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Fill in each blanks so that the resulting statement is true. Any set of ordered pairs is called a/an _______. T...

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For the following exercises, find the distance between the two points. (-12, -3) (-1, 5)

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Fill in each blanks so that the resulting statement is true. Any set of ordered pairs is called a/an _______. T...

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In the following exercises, model the subtraction. — 4

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