Find the equation of the pair of straight lines passing through the point $(1, 3)$ and perpendicular to the lines $2 x - 3 y + 1 = 0$ and $5 x + y - 3 = 0$

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Solution

Given Point is $(1, 3)$
$L_{1} = 2 x - 3 y + 1 = 0$
$L_{2} = 5 x + y - 3 = 0$
From $L_{1}$
$2 x + 1 = 3 y$
$y = \frac{2}{3} x + 1$
$∴ m_{1} (s l o p e o f L_{1}) = \frac{2}{3}$
From $L_{2}$
$y = 3 - 5 x$
$∴ m_{2} (s l o p e o f l i n e L_{2}) = - 5$
Equations of lines passing through $P (1, 3)$ and perpendicular to both $L_{1}$ and $L_{2}$ are
$(y - 3) = \frac{- 3}{2} (x - 1)$
$\Rightarrow 2 y - 6 = - 3 x + 3$
$\Rightarrow 2 y + 3 x = 9$

and
$(y - 3) = \frac{1}{5} (x - 1)$
$\Rightarrow 5 y - 15 = x - 1$
$\Rightarrow 5 y - x = 14$
Combined equation
$\Rightarrow (2 y + 3 x - 9) (5 y - x - 14) = 0$ .

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