3D Geometry 2 Question 4
4. If the line, $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane, $l x+m y-z=9$, then $l^{2}+m^{2}$ is equal to
(2016 Main)
(a) 26
(b) 18
(c) 5
(d) 2
Show Answer
Answer:
Correct Answer: 4. (d)
Solution:
- Since, the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane $l x+m y-z=9$, therefore we have $2 l-m-3=0$
$[\because$ normal will be perpendicular to the line] $\Rightarrow \quad 2 l-m=3$ …..(i)
and
$ 3 l-2 m+4=9 $
$ [\because \text { point }(3,-2,-4) \text { lies on the plane }] $
$ 3 l-2 m=5 …..(ii) $
On solving Eqs. (i) and (ii), we get
$ \begin{gathered} l=1 \text { and } m=-1 \\ l^{2}+m^{2}=2 \end{gathered} $
