2022 DHS J2 MYE P1 Q10
Timothy Gan
2022 DHS J2 MYE P1 Q10
The plane ${{\pi }_{1}}$ contains the point $A\left( 2,1,-1 \right)$ and the line $l$ with equation $\frac{x-5}{2}=y$, $z=1$. Another plane ${{\pi }_{2}}$ meets ${{\pi }_{1}}$ in the line $l$. It is known that $\left( \begin{matrix}
-1 \\
2 \\
-6 \\
\end{matrix} \right)$ is a normal for ${{\pi }_{2}}$.
(a)
Show that the equation of ${{\pi }_{1}}$ in scalar product form is $\mathbf{r}.\left( \begin{matrix}
-2 \\
4 \\
5 \\
\end{matrix} \right)=-5$.
[3]
(b)
Find the position vector of the foot of perpendicular, $F$ from $A$ to the line $l$.
Hence or otherwise, find the perpendicular distance from $A$ to ${{\pi }_{2}}$.
[3]
(c)
Find the acute angle between ${{\pi }_{1}}$ and ${{\pi }_{2}}$.
[2]
(d)
$A’$ is the point of reflection of $A$ about the plane ${{\pi }_{2}}$. Deduce the area of the triangle $AFA’$, leaving your answer in the form $\frac{a\sqrt{5}}{b}$, where $a$ and $b$ are integers to be determined.
[3]
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