Question

The coordinates of the image of the point (2, 3) in the line mirror x + y – 11 = 0 are
(a) (5, 6)
(b) (9, 8)
(c) (8, 9)
(d) (–8, –9)

Answer 1

Let us suppose A' denotes the image of (2, 3) for line mirror x + y − 11 = 0

Since coordinates of B are (5, 6)
For given line say CD; x + y – 11 = 0
Figure
Slope is given by $\frac{-1}{1}=-1$
Since AB is perpendicular to CD
Slope AB = $\frac{-1}{\mathrm{slope}\mathrm{of}CD}$
Slope of AB = 1
∴ equation of AB is y – 3 = + 1(x – 2)
i.e. y – 3 = x – 2
i.e. x – y + 1 = 0
∴ Coordinates of B is given by point of intersection of AB and CD.
i.e. x + y – 11 = 0
$$\begin{gathered} \frac{x-y+1=0}{2x-10=0} \\ x=5 \\ y=6 \end{gathered}$$
i.e. Coordinates of foot of perpendicular from (2, 3) to x + y – 11 = 0 is (5, 6)
Let A' be given by (a, b) then clearly B is midpoint of A and A'
∴ By midpoint formula,
$5=\frac{2+a}{2}\mathrm{and}6=\frac{3+b}{2}$
i.e. 10 = 2 + a and 12 = 3 + b
i.e. a = 8 and b = 9
∴ co-ordinate of the image of the point (2, 3) for line mirror x + y − 11 = 0 is (8, 9)

Hence, the correct answer is option C.