If the line joining $A(4,1,2)$ and $B(0, k, 1)$ is perpendicular to the line joining $C(-2,1,1)$ and $D(4,2…
If the line joining $A(4,1,2)$ and $B(0, k, 1)$ is perpendicular to the line joining $C(-2,1,1)$ and $D(4,2,5)$, then the value of $k$ is equal to
$31$
$-29$
$-31$
$29$
Solution
Since, we know that the direction ratio of line joining points $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ is
$ < x_2-x_1, y_2-y_1, z_2-z_1>$
Now, $A=(4,1,2), B=(0, k, 1)$
$\therefore$ Direction ratio of line $A B= < -4, k-1,-1>$
Let $a_1=-4, b_1=k-1, c_1=-1$
$C=(-2,1,1), D=(4,2,5)$
$\therefore$ Direction ratio of line $C D= < 6,1,4>$
$\Rightarrow \quad a_2=6, b_2=1, c_2=4$
Given that, lines $A B$ and $C D$ are perpendicular to each other
$\begin{array}{rrr}\Rightarrow & a_1 a_2+b_1 b_2+c_1 c_2=0 \\ \Rightarrow & (-4)(6)+(k-1)(1)+(-1)(4)=0\end{array}$
$\Rightarrow \quad-24+k-1-4=0$
$\Rightarrow \quad k=29$