The line $l$ passes through the points $A$ and $B$ with coordinates $\left( \mathrm{5, 2, 4} \right)

Question

The line $l$ passes through the points $A$ and $B$ with coordinates $\left( \mathrm{5, 2, 4} \right)$ and $\left( 4,\,-1,\,3 \right)$respectively. The plane $p$ has equation $4x+7y+5z=24$. The point $C$ lies on $l$ such that the foot of perpendicular of $C$ onto $p$ has coordinates $\left( 3,\mathrm{ }1,\mathrm{ }1 \right)$. Find the coordinates of $C$. [4] Plane ${{p}_{1}}$ has equation $3x-2y+\lambda z=\mu $. What can be said about the values of $\lambda $ and $\mu $ if $l$ does not intersect ${{p}_{1}}$? [2] Hence find the exact values of $\mu $ if the distance between ${{p}_{1}}$ and $l$ is $2$ units. [3]

Accepted Answer

(i) $C=(7,8,6)$ (ii) $\lambda=3$, $\mu\ne23$ (iii) $\mu=23\pm2\sqrt{22}$

2018 NYJC P1 Q5

2018 NYJC P1 Q5

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