Question

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

Answer 1

Line 1.

$P.(1,2)$

$m\times \left(\frac{-3}{2}\right)=-1$

$m=\frac{2}{3}$

$y=\frac{2}{3}x+C$

$\Rightarrow 2=\frac{2}{3}+C$

$\Rightarrow \frac{4}{3}=C$

Hence, $y=\frac{2}{3}x+\frac{4}{3}$

$2x-3y+4=0$

Line 2.

$P(1,2)$

$m\times \left(\frac{2}{5}\right)=-1$

$m=\frac{-5}{2}$

$y=\frac{-5x}{2}+C$

$2=\frac{-5}{2}+C$

$C=\frac{9}{2}$

$y=\frac{-5}{2}x+\frac{9}{2}$

$2y+5x-9=0$

$(2x-3y+4)(2y+5x-9)=0$

Hence, this is the answer.