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

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.