Question

Let $Q$ and $R$ be two points on the line $\frac{ x +1}{2}=\frac{ y +2}{3}=\frac{ z -1}{2}$ at a distance $\sqrt{26}$ from the point $P (4,2,7)$. Then the square of the area of the triangle $PQR$ is _______.

Answer 1

Let $(2 \lambda-1,3 \lambda-2,2 \lambda+1)$ be any point on the line
$ (2 \lambda-5)^2+(3 \lambda-4)^2+(2 \lambda-6)^2=26 $
$ \lambda=1,3 $
$ Q (1,1,3) ; R (5,7,7) ; P (4,2,7) $
Area of triangle $ PQR =1 / 2|\overrightarrow{ PQ } \times \overrightarrow{ PR }|$
$ =\sqrt{153}$