2022 JPJC J2 MYE P1 Q8
The plane ${{p}_{1}}$ has equation $2x-y+3z=0$ and the line $l$ has equation $x=\frac{y-1}{a}=\frac{z-3}{b}$, where $a$ and $b$ are non-zero constants. The points $A$ and $B$ have position vectors $9\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ and $\mathbf{j}+3\mathbf{k}$ respectively.
(i)
Given that line $l$ is parallel to plane ${{p}_{1}}$, express $a$ in terms of $b$.
[3]
(ii)
Find the position vector of the foot of perpendicular from $A$ to plane ${{p}_{1}}$ and hence find the shortest distance from $A$ to plane ${{p}_{1}}$.
[5]
(iii)
Find the length of projection of $\overrightarrow{OB}$ on the plane ${{p}_{1}}$.
[3]
Another plane ${{p}_{2}}$ has equation $cx-y+3z=d$.
(iv)
Given that $c=3$ and $d=-1$, find a vector equation of the line of intersection between ${{p}_{1}}$ and ${{p}_{2}}$.
[2]
(v)
Find the value of $c$ and the range of values of $d$ such that planes ${{p}_{1}}$ and ${{p}_{2}}$ are parallel and non-intersecting.
[2]
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