The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations \[\pi_1 : \mathbf

Question

The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations $$\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \ -3 \ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,$$ where $k$ is a constant. Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \ 0 \ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \ 1 \ -1 \end{pmatrix}, \beta \in \mathbb{R}$. Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.[4] Determine the possible value(s) of $k$ such that $l_1$ and $l_2$ are skew.[3] Assume that $k = 4$ for the rest of this question. Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that $\overrightarrow{AB}$ is perpendicular to both lines. Show that $\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where $p$ is an integer to be determined.[3] Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.[2] It is given further that $l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$. Deduce the shortest distance from $P$ to $l_1$.[2]

Accepted Answer

(i) $\mathbf r=\begin{pmatrix}7\0\11\end{pmatrix}+\lambda\begin{pmatrix}6-k\5\2k+3\end{pmatrix}$ (ii) $k\ne-5$ (iii) $p=297$ (iv) $\dfrac2{\sqrt{154}}$ (v) $\dfrac{33\sqrt{21}}2$

2024 NJC P1 Q12

2024 NJC P1 Q12

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