If the line joining the points $A(7, p, 2)$ and $B(q,-2,5)$ is parallel to the line joining the points $C(2…
If the line joining the points $A(7, p, 2)$ and $B(q,-2,5)$ is parallel to the line joining the points $C(2,-3,5)$ and $D(-6,-15,11)$, then the value of $p^2+q^2$ is equal to
25
16
9
7
Solution
$
\begin{aligned}
& \text { (a) } A(7, p, 2), B(q,-2,5) \\
& C(2,-3,5), D(-6,-15,11)
\end{aligned}
$
Direction ratios of $A B$ is given as,
$
\begin{aligned}
& q-7,-2-p, 5-2 \\
& \Rightarrow a_1=q-7, b_1=-2-p, c_1=3
\end{aligned}
$
Direction ratios of $C D$ is given as,
$
a_2=-8, b_2=-12, c_2=6
$
Given that, $A B \| C D \Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
$
\begin{aligned}
& \Rightarrow & \frac{q-7}{-8} & =\frac{-2-p}{-12}=\frac{3}{6} \\
\Rightarrow & & \frac{q-7}{-8} & =\frac{3}{6} \\
\Rightarrow & & q-7 & =-4 \\
\Rightarrow & & q & =3 \\
& \text { and } & \frac{-2-p}{-12} & =\frac{3}{6}
\end{aligned}
$
$
\begin{array}{lrl}
\Rightarrow & p+2 & =6 \Rightarrow p=4 \\
\therefore & p^2+q^2 & =16+9=25
\end{array}
$