Question

The reflection of the point \[\left( {6,8} \right)\] in the line $x = y$ is
A) $\left( {4,2} \right)$
B) $\left( { - 6, - 8} \right)$
C) \[\left( { - 8, - 10} \right)\]
D) \[\left( {8,6} \right)\]

Answer 1

Hint:
Find the coordinates of the midpoint of the given point and the point of reflection. Substitute the value of mid-point in the given equation of line $x = y$. Now, the product of the slope of line joining the points and the given line will be \[ - 1\]. Use this to find another equation in terms of coordinates of reflection. Solve the equations to find the coordinates of reflection.

Complete step by step solution:
Let the line $x = y$be \[AB\]and the point \[P\left( {6,8} \right)\].
Let \[Q\left( {h,k} \right)\] be the point of reflection of \[\left( {6,8} \right)\] in the line $x = y$
Line \[AB\]acts like a mirror.
Then, \[P\left( {6,8} \right)\] and \[Q\left( {h,k} \right)\] are at equal distances from the line $x = y$
Let \[R\]be the mid-point of the line \[AB\].
Then coordinates of \[R\] are \[\left( {\dfrac{{h + 6}}{2},\dfrac{{k + 8}}{2}} \right)\] from the mid-point formula.
Also, \[R\left( {\dfrac{{h + 6}}{2},\dfrac{{k + 8}}{2}} \right)\] lies on the line $x = y$, it will satisfy the equation. This implies,
$
  \dfrac{{h + 6}}{2} = \dfrac{{k + 8}}{2} \\
  h + 6 = k + 8 \\
  h - k = 2{\text{ }}\left( 1 \right) \\
$
Also, \[PQ \bot AB\], therefore, the product of slope of \[PQ\] and \[AB\]is equals to \[ - 1\].
Slope of \[AB\]=\[ - \dfrac{{{\text{coefficient of }}x}}{{{\text{coefficient of }}y}}\] which is 1
Hence, slope of \[PQ\] is \[ - 1\] which is also equals to \[\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{k - 8}}{{h - 6}}\]
$
  \dfrac{{k - 8}}{{h - 6}} = - 1 \\
  k - 8 = - h + 6 \\
  h + k = 14{\text{ }}\left( 2 \right) \\
$
Solve equation (1) and (2) to find the value of $\left( {h,k} \right)$
Add equation (1) and (2)
$
  h - k + h + k = 2 + 14 \\
  2h = 16 \\
  h = 8 \\
$
Substitute the value of $k$in equation (1)
$
  8 - k = 2 \\
  k = 6 \\
$
Therefore, the reflection of the point \[\left( {6,8} \right)\] in the line $x = y$ is $\left( {8,6} \right)$

Hence, option D is correct.

Note:
The line $x = y$ passes through the origin. This question can alternatively be done by using the condition that the reflection of the point $\left( {x,y} \right)$ in the line $x = y$ is $\left( {y,x} \right)$. This implies that $x$ coordinate and the $y$ coordinate interchanges.