If the line x-32=y+2-1=z+43 lies in the plane lx+my-z=9, then l^2+m^2 is equal to:

Question

If the line $\dfrac{x-3}2=\dfrac{y+2}{-1}=\dfrac{z+4}3$ lies in the plane $lx+my-z=9$, then $l^2+m^2$ is equal to:

Accepted Answer

Line lies in the plane $$ \Rightarrow \left( {3, - 2, - 4} \right)$$ lie in the plane

$$ \Rightarrow 3\ell  - 2m + 4 = 9$$ or $$3\ell  - 2m = 5....\left( 1 \right)$$

Also, $$\ell ,$$ $m, - 1$ are dr's of line perpendicular to plane

and $2, - 1,3$ are dr's of line lying in the plane

$$ \Rightarrow 2\ell  - m - 3 = 0\,\,\,$$ or $$\,\,\,2\ell  - m = 3....\left( 2 \right)$$

Solving $(1)$ and $(2)$ we get $$\ell  = 1$$ and $m=-1$

$$ \Rightarrow {\ell ^2} + {m^2} = 2.$$

Solutions · 1