Standard +0.8 This is a Further Maths question requiring multiple coordinated steps: finding the perpendicular from A to the plane using the normal vector, determining the foot of the perpendicular by solving simultaneous equations, then using the midpoint property to find the reflection. While the individual techniques are standard, the multi-step coordination and spatial reasoning required place it above average difficulty.
8 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\).
Distance between point and reflected point =4 21 M1
Reflected point is ( x,y,z )⇒( x−9 )2 +( y−6 )2 +( z−1 )2
( 9+4λ,6+2λ,1+λ)
Any point on normal line is B1
⇒16λ2 +4λ2 +λ2 =336⇒21λ2 =336⇒λ2 =16 M1
⇒λ=±4 A1
( )
25,14,5 is the same side A1
Answer
Marks
⇒(−7,−2,−3 )
B1
M1
M1
M1
A1
A1
Answer
Marks
[6]
1.1
3.1a
3.1a
1.1a
1.1
Answer
Marks
3.2a
Direction
Attempt to find point on
line
Attempt to find λ
Double λ
Answer
Marks
λ soi
Alternative methods
possible
Must be coordinates
Question 8:
8 | 4
AB has direction 2
1
and any point on it is (9+4λ, 6+2λ, 1+λ)
If this point lies on plane
then 4(9+4λ)+2(6+2λ)+(1+λ)=7
⇒49+21λ=7⇒21λ=−42⇒λ=−2
So B is where λ=−4
⇒ B has coordinates (−7,−2,−3)
Alternatively:
Distance between point and plane
42
= =2 21
21
Distance between point and reflected point =4 21 M1
Reflected point is ( x,y,z )⇒( x−9 )2 +( y−6 )2 +( z−1 )2
( 9+4λ,6+2λ,1+λ)
Any point on normal line is B1
⇒16λ2 +4λ2 +λ2 =336⇒21λ2 =336⇒λ2 =16 M1
⇒λ=±4 A1
( )
25,14,5 is the same side A1
⇒(−7,−2,−3 ) | B1
M1
M1
M1
A1
A1
[6] | 1.1
3.1a
3.1a
1.1a
1.1
3.2a | Direction
Attempt to find point on
line
Attempt to find λ
Double λ
λ soi | Alternative methods
possible
Must be coordinates
8 The equation of a plane is $4 x + 2 y + z = 7$.\\
The point $A$ has coordinates $( 9,6,1 )$ and the point $B$ is the reflection of $A$ in the plane.\\
Find the coordinates of the point $B$.
\hfill \mbox{\textit{OCR Further Pure Core 1 2019 Q8 [6]}}