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  4. If the line, (x-3/2) = (y+2/-1) = (z+4/3) lies in the plane, ℓ x + my - z = 9 , then ℓ2 + m2 is equal to :

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

JEE MainJEE Main 2016Three Dimensional Geometry

Solution:

Point on line is $P = (3, -2, -4) $
‘$P$’ lies on $\ell x + my - z = 9 $
$\Rightarrow \, 3 \ell - 2m + 4 = 9$
$3 \ell -2m = 5$ .....(1)
As line lies on plane
$\Rightarrow \, \, 2 \times \ell + m \times (-1) + 3 \times (-1) = 0$
$2 \ell - m = 3$ ....(3)
Solving $\ell = 1 , m = - 1 $
So, $\ell^2 + m^2 = 2$