The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $\left( -3,-4,11 \right)$,

Question

The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $\left( -3,-4,11 \right)$, $\left( 1,-2,0 \right)$ and $\left( -5,2,-1 \right)$ respectively. The point $M$ has position vector given by $\mathbf{m}=3\mathbf{i}+6\mathbf{j}+15\mathbf{k}$. Show that $\overrightarrow{AM}$ is perpendicular to $p$. [3] Find the coordinates of $N$ which is the mirror image of $M$ in $p$. [2] Find a vector equation of the line which is a reflection of the line $MB$ in $p$. [2] The plane $q$ has equation $11x+6y-z-4+k\left( -x-3y+z-4 \right)=0$ for some constant $k$. Given that $q$ contains $M$, find a cartesian equation of $q$. [2] By finding the line of intersection between $p$ and $q$, or otherwise, find a cartesian equation of the plane which is a reflection of $q$ in $p$. [5]

Accepted Answer

(i) shown; (ii) N(-9,-14,7); (iii) r=(1,-2,0)+λ(10,12,-7); (iv) k=5, q:6x-9y+4z=24; (v) q':-9x+4y-6z=-17

2019 NJC Promo Q11

2019 NJC Promo Q11

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