Question

If the image of the point $\left(2,1\right)$ with respect to the line mirror be $\left(5,2\right)$, then the equation of the mirror is

• A. $3x+y-12=0$
• B. $3x-y+12=0$
• C. $3x+y+12=0$
• D. none of these

Answer 1

The correct option is A

$3x+y-12=0$

Explanation for correct answer:

Using slope concept of line:

Let $CD$ be the mirror with slope $m_1$

The slope, $m$ of the line joining $A\left(2,1\right)$ and $B\left(5,2\right)$ , where $x_1=2,x_2=5,y_1=1and{y}_2=2$ will be

$$\begin{gathered} \Rightarrow m_{}=\frac{y_2-y_1}{x_2-x_1} \\ \Rightarrow m_{}=\frac{2-1}{5-2} \\ \Rightarrow m_{}=\frac{1}{3} \end{gathered}$$

Since, $CD\perp AB$

$$\begin{gathered} \therefore m_1\times m=-1 \\ \Rightarrow m_1\times \frac{1}{3}=-1 \end{gathered}$$

$\Rightarrow m_1=-3$

Using Mid point formula , $(x,y)=\left(\frac{x_1+x_2}{2},\frac{{y_{}}_1+y_2}{2}\right)$

mid point of line $AB=\left(\frac{2+5}{2},\frac{1+2}{2}\right)$$=\left(\frac{7}{2},\frac{3}{2}\right)$

Since, $\left(\frac{7}{2},\frac{3}{2}\right)$ is the mid point of line AB, therefore we can say that the mirror passes through AB.

Equation of the mirror $CD$ passing through $\left(\frac{7}{2},\frac{3}{2}\right)$ and having slope as $m_1=-3$ where $x_1=\frac{7}{2}and{y}_1=\frac{3}{2}$ , is

$y-y_1=m_1\left(x-x_1\right)$

$\Rightarrow y-\frac{3}{2}=-3\left(x-\frac{7}{2}\right)$

$\Rightarrow 2y-3=-6x+21$

$\Rightarrow 6x+2y-24=0$

$\Rightarrow 3x+y-12=0$

Therefore, option $\left(A\right)$ is correct