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$$\Rightarrow (2y+3x-9)(5y-x-14)=0$$.Maths

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