The point A has coordinates $(1,-2,4)$ and the plane $\pi_1$ has the equation $2x-y+2z=5$ Find

Question

The point A has coordinates $(1,-2,4)$ and the plane $\pi_1$ has the equation $2x-y+2z=5$ Find the exact shortest distance between $A$ and $\pi_1$.[2] The plane $\pi_2$ has equation $x+3y-az=3$, where $a$ is a constant. Find the vector equation of line $l$, which is the line of intersection of $\pi_1$ and $\pi_2$, in terms of $a$.[4] The plane $\pi_3$ has equation $bx-2y+4z=3$, where $b$ is a constant. Show that $(a-6)(b-4)=0$ if $l$ is parallel to $\pi_3$.[2] State the conditions that $a$ and $b$ must follow for the three planes to form a triangular prism, where all the planes are non-parallel and they do not have any point in common. Justify your answer.[4]

Accepted Answer

(a) $\frac73$ units. (b) $\boldsymbol r=\begin{pmatrix}\frac{18}{7}\ \frac17\ 0\end{pmatrix}+\lambda\begin{pmatrix}a-6\ 2a+2\ 7\end{pmatrix}$, $\lambda\in\mathbb R$. (c) Shown. (d) $a=6$, $b\notin\{-\frac23,\frac{23}{18},4\}$.

2025 DHS P1 Q9

2025 DHS P1 Q9

Solution:

Solution locked