Question

If the line $\frac{x+1}{-1}=\frac{y-1}{2}=\frac{z-2}{1}$ lies on the plane $nx+my-2z=4,$ then $m-n=$

Answer 1

Given line, $\frac{x+1}{-1}=\frac{y-1}{2}=\frac{z-2}{1}$ passes through point $A(-1,1,2)$
$D.r^{\prime }s$ of line is, $(a,b,c)=(-1,2,1)$
as line lies on the plane, point $A$ satisfy the plane $nx+my-2z=4,$
$\Rightarrow -n+m-4-4=0\Rightarrow -n+m=8\cdots \left(i\right)$
$d.r^{\prime }s$ of normal to the plane is : $(a_1,b_1,c_1)=(n,m,-2),$ must be perpendicular to $(a,b,c)=(-1,2,1).$
$\Rightarrow n(-1)+m\left(2\right)-2\left(1\right)=0$
$\Rightarrow -n+2m=2\cdots \left(ii\right)$
Solving $\left(i\right)$ and $\left(ii\right),$ we get
$m=-6,n=-14$