Question

Question 24 A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

Class X1 - Maths -Straight Lines Page 234

Answer 1

Equations of straight path are
2x - 3y + 4 = 0 ____________(1)
and , 3x + 4y - 5 = 0 ________(2)
solve equations (1) and (2) ,

eqn (1) × 4 +eqn (2) × 3

(2x -3y + 4) × 4 + (3x + 4y - 5) × 3 = 0
8x - 12y + 16 + 9x + 12y - 15 = 0
17x + 1 = 0
x = -1/17 and put it in equation (1)
2(-1/17) -3y + 4 = 0
3y = - 2/17 + 4 = 66/17
y = 22/17

suppose the person standing at P , then he can reach the path AB in least time if he goes straight to it. I mean perpendicular to it .

it means we have to find the equation of PQ

Line PQ ⊥ line AB
so,. slope of PQ × slope of AB = -1
slope of PQ × (6/7) = -1 [ slope of 6 -7y+ 8 is (6/7) ]

slope of PQ = -7/6

now, equation of line PQ where , P(-1/17,22/17) and slope of line is -7/6 :
(y - 22/17) = (-7/6)(x +1/17)
6(y-22/17) + 7(x + 1/17) = 0
6 × 17 y - 22 × 6 + 7 × 17 x + 7 = 0
102y -132 + 102x + 7 = 0
102x + 102y - 125 = 0

Answer 2

Hey !!

The equation of the given line are

2x - 3y + 4 = 0  -------> (1)

3x + 4y - 5 = 0  --------> (2)

6x - 7y + 8 = 0  ---------> (3)

The person is standing at the junction of the paths represented by lines (1) and (2)

Now, on solving equations (1) and (2) we obtain,

x = $\frac{-1}{7}$ and y = $\frac{22}{17}$

Thus, the person is standing at point $(-\frac{1}{17} , \frac{22}{17}$)

The person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point

$(-\frac{1}{17} , \frac{22}{17} )$ Slope of the line (3) = $\frac{6}{7}$

∴ Slope of the line perpendicular to line (3)

= -1 / 6/7 = $(-\frac{7}{6} )$

The equation of the line passing through

$(-\frac{1}{17} , \frac{22}{17} )$ and having a slope of $(-\frac{7}{6} )$

( y-22/17) = -7/6 (x + 1/17)

= 6(17 y - 22) = -7(x + 1/17)

= 102y - 132 = -119x - 7

= 119x + 102y = 125

Hence, the path that the person should follow is

FINAL RESULT = 119x + 102y = 125

GOOD LUCK !!