Question

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.

Answer 1

The equations of the given lines 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). On solving equations (1) and (2), we obtain $x = -\frac{ 1}{17}$ and $y=\frac{22}{7}$

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

The equation of the line passing through $(\frac{-1}{17}, \frac{22}{17})$ and having a slope of $\frac{-7}{6}$ is given by

$(y-\frac{22}{17}) =\frac{-7}{6}(x+\frac{1}{17})$

$6 (17y – 22) = – 7 (17x + 1)$
$102y – 132 = – 119x – 7$
$1119x + 102y = 125$

Hence, the path that the person should follow is $119x + 102y = 125.$