Question

Find the equations to the straight lines that pass through the intersection of the lines $2x-3y=0$ and $4x-5y=2$ and
$\left(i\right)$ Passes through the point $(2,1)$.
$\left(ii\right)$ Is perpendicular to the line $x+2y+1=0$.
$\left(iii\right)$ Is parallel to the line $3x-4y+5=0$.

Answer 1

Any line through the intersection of given lines is $P+\lambda Q=0$
or $2x-3y+\lambda (4x-5y-2)=0$
or $x(2+4\lambda )-y(3+5\lambda )-2\lambda =0$
(i) Pass through $(2,1)\Rightarrow \lambda =-1$.
(ii) $\perp$ to $x+2y+1=0$
$\therefore a_1{a}_2+b_1{b}_2=0\Rightarrow \lambda =-2/3$.
(iii) Parallel to $3x-4y+5=0$
$\therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}\Rightarrow \lambda =1$
Putting the values of $\lambda$ in $\left(1\right)$, the required equations are
(i) $x-y-1=0$
(ii) $2x-y-4=0$
(iii) $3x-4y-1=0$.