Question

A traffic policeman at (3, 5) wants to reach the nearest point on the road along the line 5 + 6 = 106 as early as possible to catch a defaulter. Locate the position of .

Answer 1

Given : A traffic policeman at P (3, 5) wants to reach the nearest point Q on the road along the line 5x + 6y = 106 as early as possible to catch a defaulter.

To Find : Locate the position of Q

Find the distance PQ

Solution:

5x + 6y = 106

y = -5x/6 + 106/6

Slope =  - 5/6

Perpendicular Distance will be Shortest

Hence PQ ⊥  5x + 6y = 106

Slope of PQ  = 6/5      

P (3, 5)  

Equation of PQ  

y - 5  = (6/5) (x - 3)

=> 5y - 25 = 6x - 18

=> 6x  - 5y   = - 7

5x + 6y = 106

 6x  - 5y   = - 7

on solving

x = 8  , y = 11

Hence  Q = ( 8 , 11)

Q is located at (8 , 11)

P (3, 5)   , Q ( 8 , 11)

Distance = √(8 - 3)² + (11 - 5)²  = √5² + 6² = √61 = 7.81

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Answer 2

Step-by-step explanation:

Given A traffic policeman at (3, 5) wants to reach the nearest point on the road along the line 5x + 6y = 106 as early as possible to catch a defaulter. Locate the position of Q.

• So after drawing a graph we have the equation
•         5x + 6y = 106
• Substituting y = 0 in x intercept we have  
•                 x = 106 / 5
•              x = 21.2
•      if x = 0 we have  
•         y = 106 / 6
•        y = 17.67
• So we get (21.2 , 0) and (0, 17.67)
• Now we get a straight line.
• Substituting (3,5) in the given equation we get
•       5x + 6y = 5(3) + 6(5)
•                    = 15 + 30
•                    = 45
• Now 45 < 106
• So point P (3,5) will be in the middle.
• We need to find the point of shortest distance.
• So perpendicular distance PQ will be
•      PQ = 5 x + 6y
•           = 5 x 3 + 6 x 5 – 106 / √5^2 + 6^2
•          =  mod 15 + 30 – 106 / √61
•             = 61 / √61  
•              = √61
• So now slope of equation 5x + 6y = 106 will be m = - a/b
•                                                                       m = - 5 / 6
•             So slope m of PQ = 6/5 (Since PQ is perpendicular to line L)
•                 So to find in coordinates we have
•                     Now β – 5 / α – 3 = 6/5
•                          5β – 25 = 6α – 18
•                          5β - 6α = 7 ------------1 x by 5
•                   Also 5α + 6β = 106 --------2 x by 6
•                           25β - 30α = 35
•                           36 β +30α = 636
•                                 61 β = 671
•                               β = 671 / 61
•                                β = 11
•                           So 5 (11) - 6α = 7
•                                  55 – 7 = 6α
•                                   α = 48 / 6
•                             Or α = 8
• Therefore the coordinates are (8, 11)

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https://brainly.in/question/28623621