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Question

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


Solution

Given Point is $$(1, 3)$$ 
$$L_1=2x-3y+1=0$$
$$L_2=5x+y-3=0$$
From $$L_1$$
$$2x+1=3y$$
$$y=\dfrac{2}{3}x+1$$
$$\therefore m_1(slope\space of\space L_1)=\dfrac{2}{3}$$
From $$L_2$$
$$y=3-5x$$
$$\therefore m_2(slope\space of\space line\space L_2)=-5$$
Equations of lines passing through $$P(1, 3)$$ and perpendicular to both $$L_1 $$ and $$L_2$$ are
 $$ (y-3)=\dfrac{-3}{2}(x-1)$$
$$\Rightarrow 2y-6=-3x+3$$
$$\Rightarrow 2y+3x=9$$

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

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