Question

The position of reflection of the point $\left(4,1\right)$ about the line$y=x-1$ is

• A. $\left(1,2\right)$
• B. $\left(3,4\right)$
• C. $\left(-1,0\right)$
• D. $\left(2,3\right)$

Answer 1

The correct option is D

$\left(2,3\right)$

Solution:

Step 1: Given data:

The position of reflection of the point $=\left(4,1\right)$

The line$y=x-1$

Step 2: Draw the diagram:

Suppose that$Q$ be the image of $P$. Join$PQ$ and it cuts the line mirror$y=x-1$at point $R$.

Note that $R=\left(h,k\right)$ and $R$ lies on $y=x-1$.

Therefore $k=h-1......................\left(1\right)$

Step 3: Formula used:

The slope between

$PR=\frac{y_2-y_1}{x_2-x_1}$

Step 4: Now calculate the slope:

Slope $PR=\frac{k-1}{h-4}$

From the equation (1).

$PR=\frac{h-2}{h-4}$

Note that the line mirror is perpendicular to PR therefore the slope of $PR=-1$.

Then,

$\begin{array}{rcl}-1& =& \frac{h-2}{h-4}\\ -h+4& =& h-2\\ 2h& =& 6\\ h& =& 3\end{array}$

Substitute the value of $\begin{array}{rcl}& & h\end{array}$ in equation (1).

Therefore,

$$\begin{gathered} k=3-1 \\ k=2 \end{gathered}$$

Therefore the point of $R=\left(3,2\right)$

Know that $PR=RQ$where $R$is the midpoint.

Suppose that $Q=\left(a,b\right)$

Then the coordinates of $Q$

$\begin{array}{rcl}\frac{4+a}{2}& =& 3\\ a& =& 2\\ \frac{1+b}{2}& =& 2\\ b& =& 3\end{array}$

Therefore the coordinates of $Q=\left(2,3\right)$.

Hence, the correct option is (D).