2022 TMJC P1 Q8
The plane ${{p}_{1}}$ has cartesian equation $x+z=3$. The plane ${{p}_{2}}$ is perpendicular to ${{p}_{1}}$ and contains the line ${{l}_{1}}$ with equation $\frac{x+2}{5}=\frac{y+1}{2}=\frac{3-z}{3}$.
(i)
Show that the cartesian equation of the plane ${{p}_{2}}$ is $-x+4y+z=1$.
[2]
(ii)
Find a vector equation of the line ${{l}_{2}}$ given that ${{p}_{1}}$ and ${{p}_{2}}$ intersect at ${{l}_{2}}$.
[2]
(iii)
It is given that the point $B$ with coordinates $\left( 0,4,3 \right)$ is on ${{p}_{1}}$ and the perpendicular distance from $B$ to ${{p}_{2}}$ is $k$. Find the position vector of the foot of perpendicular from $B$ to ${{p}_{2}}$ and deduce the value of $k$.
[4]
(iv)
Hence, find the vector equations of the lines in ${{p}_{1}}$ such that the perpendicular distance from each line to ${{p}_{2}}$ is $k$.
[3]
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