Question

Obtain the equations of the lines passing through the intersection of lines $4x-3y-1=0$ and $2x-5y+3=0$ and equally inclined to the axes

Answer 1

The equation of any line through the intersection of the given lines is

$(4x-3y-1)+\lambda (2x-5y+3)=0$

or $x(2\lambda +4)-y(5\lambda +3)+3\lambda -1=0$ ........(i)

Let m be the slope of this line Then m = $\frac{2\lambda +4}{5\lambda +3}$

As the line is equally inclined with the axes therefore

m = $\tan {45}^{\circ }orm=\tan {135}^{\circ }$

$m=\pm 1,\frac{2\lambda +4}{5\lambda +3}=\pm 1$

$\Rightarrow \lambda =-1or\frac{1}{3}$ putting the values of $\lambda$ in (i) we get 2x + 2y - 4 = 0 and 14x - 14y = 0

i.e. x + y - 2 = 0 and x = y as the equations of the required lines