Let A and B be two distinct points on the line $\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Both A and B are at a distance $2 \sqrt{17}$ from the foot of perpendicular drawn from the point $(1,2,3)$ on the line L . If O is the origin, then $\overrightarrow{O…

Question

Let A and B be two distinct points on the line $\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Both A and B are at a distance $2 \sqrt{17}$ from the foot of perpendicular drawn from the point $(1,2,3)$ on the line L . If O is the origin, then $\overrightarrow{O A} \cdot \overrightarrow{O B}$ is equal to:

Aicharya · Accepted Answer

$\begin{aligned}
& \overrightarrow{\mathrm{PQ}} \cdot \overrightarrow{\mathrm{~b}}=0 \\ & \Rightarrow 3(3 \lambda+5)+2(2 \lambda+5)-2(-2 \lambda+4) \\ & \Rightarrow 17 \lambda=-17 \Rightarrow \lambda=-1 \\ & \mathrm{Q}(3,5,9)
\end{aligned}$
Let A $(3 \mu+6,2 \mu+7,-2 \mu+7)$
$(3 \mu+3)^2+(2 \mu+2)^2+(-2 \mu-2)^2=68$
$\Rightarrow \mu^2+2 \mu-3=0 \mu=-3 \text { or } \mu=1$
A $(-3,1,13)$ and $B(9,9,5)$
$\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=-27+9+65=47$ *

Let A and B be two distinct points on the line $\mathrm{L}:…

Solution