Let P be the image of the point $Q(7,-2,5)$ in the line $L:\frac{x-1}2=\frac{y+1}3=\frac z4$ and $R(5,p,q)$ be a point on $L$. Then the square of the area of $ riangle PQR$ is ________.

Question

Let P be the image of the point $Q(7,-2,5)$ in the line $L:\frac{x-1}2=\frac{y+1}3=\frac z4$ and $R(5,p,q)$ be a point on $L$. Then the square of the area of $	riangle PQR$ is ________.

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] A point of $L$ is $(1+2t,-1+3t,4t)$. The foot $F$ of the perpendicular from $Q$: $(2t-6)2+(3t+1)3+(4t-5)4=0\Rightarrow t=1$, $F=(3,2,4)$ and $P=2F-Q=(-1,6,3)$. $R$ on $L$ with $x=5$ gives $t=2$: $R=(5,5,8)$. $\vec{PQ}=(8,-8,2)$, $\vec{PR}=(6,-1,5)$, $\vec{PQ}	imes\vec{PR}=(-38,-28,40)$ with $|\cdot|^2=3828$. Area$^2=\frac{3828}4=957$. Answer: 957.

Q. Let P be the image of the point $Q(7,-2,5)$ in the line $L:\frac{x-1}2=\frac{y+1}3=\frac z4$ and $R(5,p,q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.

Explanation

Reviewed by Dr. Abubacker

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