Question

Let the lines
$L_1:\vec r=λ(\hat i+2\hat j+3 \hat k), \ λ∈R$
$L_2:\vec r=(\hat i+3\hat j+\hat k)+μ(\hat i+\hat j+\hat 5k), \ μ∈R$
intersect at the point $S$. If a plane $ax + by – z + d = 0$ passes through $S$ and is parallel to both the lines $L_1$ and $L_2$ then the value of $a + b + d$ is equal to _______.

Answer 1

As plane is parallel to both the lines we have d.r’s of normal to the plane as $(7, – 2, –1)$ from $\begin{vmatrix} \hat i & \hat i & \hat k\\\ 1 & 2 & 3 \\\ 1 & 1 &5 \end{vmatrix} = 7\hat i−\hat j(2)+\hat k(−1)$
The point of intersection of lines is $2\hat i+4\hat j+6\hat k$
So, the equation of plane is,
$7(x – 2) –2 (y – 4) – 1 (z – 6) = 0$
$7x – 2y – z = 0$
$a + b + d = 7 – 2 + 0 = 5$
$a + b + d = 5$

So, the answer is $5$.