2021 TMJC P1 Q9
The plane ${{\pi }_{1}}$ contains the point $B$ with coordinates $\left( 2,-1,3 \right)$ and the line $l$.
(i)
Show that the cartesian equation of ${{\pi }_{1}}$ is $x-y+4z=15$.
[2]
Let $F$ be the point on $l$ which is closest to $B$.
(ii)
Find the coordinates of $F$.
[3]
Point $C$ has coordinates $\left( 8,1,-3 \right)$.
(iii)
Find the exact shortest distance from $C$ to ${{\pi }_{1}}$.
[2]
(iv)
Hence, find the exact volume of the tetrahedron $CABF$.
[3]
[Volume of tetrahedron$=\frac{1}{3}\times $base area$\times $perpendicular height]
(v)
$D$ is a general point on plane ${{\pi }_{2}}$ such that $DABF$ has the same volume as tetrahedron $CABF$. Given that ${{\pi }_{2}}$ does not contain $C$, find an equation of ${{\pi }_{2}}$.
[2]
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