Intersections and distances involving planesAQA A-Level Further Maths: Subtopic test

1. 1
   The plane $\Pi$ has equation $2x-y+2z=9$ and the point $A$ has coordinates $(1,2,0)$.

   (a)

   Find the perpendicular distance from $A$ to $\Pi$.

   [1 mark]

   • A$1$
   • B$9$
   • C$\frac95$
   • D$3$

   (b)

   Find the coordinates of the point on $\Pi$ closest to $A$.

   [1 mark]

   • A$(3,1,2)$
   • B$(-1,3,-2)$
   • C$(7,-1,6)$
   • D$(2,-1,2)$

   (c)

   Find the coordinates of the image of $A$ when it is reflected in $\Pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l$ has equation $\mathbf r=\begin{pmatrix}1 \\ -2 \\ 3\end{pmatrix}+t\begin{pmatrix}2 \\ 1 \\ -1\end{pmatrix}$ and the plane $\Pi$ has equation $3x-y+2z=17$.

   (a)

   Find the value of $t$ at the point where $l$ meets $\Pi$.

   [1 mark]

   • A$-2$
   • B$6$
   • C$2$
   • D$\frac{28}{3}$

   (b)

   Find the coordinates of the point where $l$ meets $\Pi$.

   [1 mark]

   • A$(-3,-4,5)$
   • B$(5,0,1)$
   • C$(4,2,-2)$
   • D$(5,0,5)$

   (c)

   The line $m$ has equation $\mathbf r=\begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}+s\begin{pmatrix}1 \\ -1 \\ -2\end{pmatrix}$. Show that $m$ does not meet $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $L$ has equation $\dfrac{x-1}{2}=\dfrac{y+3}{1}=\dfrac{z-2}{-3}$ and the plane $\Pi$ has equation $x-2y+z=3$.

   (a)

   Find the coordinates of the point where $L$ meets $\Pi$.

   [3 marks]

   (b)

   The point $P(1,-3,2)$ lies on $L$. Find the perpendicular distance from $P$ to $\Pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone starts at the point $A(-2,4,10)$ and flies in a straight line with direction $\begin{pmatrix}1 \\ -2 \\ -3\end{pmatrix}$ towards a flat roof. Taking one unit as one metre, the roof lies in the plane $2x+3y+z=4$.

   (a)

   (i) Find the coordinates of the point where the path of the drone meets the roof.
   (ii) Find the distance the drone travels from $A$ to the roof.

   [6 marks]

   (b)

   Find the shortest distance from $A$ to the roof, and the coordinates of the point on the roof nearest to $A$.

   [6 marks]

   Total for question 4: 12 marks

End of questions