Question

Reduce in symmetrical form, the equations of the line $x=ay+b,z=cy+d$.

Answer 1

Let $l,m,n$ be the direction ratios of the required line. Since the required line lies in both the given planes,

$l+m(-a)+n.0=0$ and $l.0+m(-c)+n=0$

Solving these two equations, we get,

$\frac{l}{-a}=\frac{m}{-1}=\frac{n}{-c}$ or $\frac{l}{a}=\frac{m}{1}=\frac{n}{c}$

Putting $y=0$ in the two equations, we get,

$x=b,z=d$

So, the coordinates of a point on the required lines are $(b,0,d)$.

Therefore, equation is

$\frac{x-b}{a}=\frac{y-0}{1}=\frac{z-d}{c}$