Question

The image of the point$p\left( {4, - 13} \right)$ in the line mirror$5x + y + 6 = 0$, is:
A. $\left( {1, - 14} \right)$
B. $\left( {6, - 15} \right)$
C. $\left( { - 1, - 14} \right)$
D. $\left( {1,14} \right)$

Answer 1

This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. Also, we need to know how to solve two equations to make easy calculations. We need to know the formula with the involvement of two points with slope value to solve these types of questions.

Complete step by step solution:
In this question we have,
The image of the point is $p\left( {4, - 13} \right)$ in the line mirror $5x + y + 6 = 0$.
The above-mentioned equation can also be written as,
$5x + y + 6 = 0$
$5x + 6 = - y$
$y = - 5x - 6 \to \left( 1 \right)$
Now the equation $\left( 1 \right)$ is in the form of $y = mx + c$
So, the slope ${m_1} = - 5$
The slope of the perpendicular line with the given line is,
${m_2} = \dfrac{{ - 1}}{{{m_1}}} = \dfrac{{ - 1}}{{ - 5}}$
${m_2} = \dfrac{1}{5}$
The point given in the question is, $\left( {4, - 13} \right)$
We know that,
$y - {y_1} = m\left( {x - {x_1}} \right)$
Here,$\left( {{x_1},{y_1}} \right) = \left( {4, - 13} \right)$, take $m = {m_2} = \dfrac{1}{5}$
So, we get

$$y + 13 = \dfrac{1}{5}\left( {x - 4} \right) \\\ y + 3 = \dfrac{x}{5} - \dfrac{4}{5} \\\$$

By solving the above equation we get,

$$y = \dfrac{x}{5} - \dfrac{4}{5} - 13 \\\ y = \dfrac{x}{5} - \left( {\dfrac{4}{5} + 13} \right) \\\$$

Let’s convert the mixed fraction into a simple fraction. So, we get
$y = \dfrac{1}{5}x - 13.8 \to \left( 2 \right)$
Let’s solve the equation $\left( 1 \right)\& \left( 2 \right)$
$\left( 1 \right) \to y = - 5x - 6$
$\left( 2 \right) \to y = \dfrac{1}{5}x - 13.8$
Let’s subtract the equation $\left( 2 \right)$ and $\left( 1 \right)$ as given below,

$$\left( 2 \right) \to y = \dfrac{1}{5}x - 13.8 \\\ \- \left( 1 \right) \to - y = + 5x + 6 \\\ \Rightarrow 0 = \dfrac{{26}}{5}x - 7.8 \to \left( 3 \right) \\\$$

Here$\left( {\dfrac{1}{5} + 5 = \dfrac{{25 + 1}}{5} = \dfrac{{26}}{5}} \right)$
So, the equation$\left( 3 \right)$becomes,

$$\dfrac{{26}}{5}x = 7.8 \\\ x = 7.8 \times \dfrac{{26}}{5} \\\$$

$x = 1.5$
Let’s substitute $x = 1.5$ in the equation $\left( 1 \right)$, we get

$$y = - 5\left( {1.5} \right) - 6 \\\ y = - 7.5 - 6 \\\ y = - 13.5 \\\$$

So, we get the intersection point $\left( {1.5, - 13.5} \right)$ is the midpoint of $\left( {4, - 13} \right)$and its image say $\left( {p,q} \right)$
Let’s find the value $p$. We know that,
${x_1} = \dfrac{{{x_2} + p}}{2}$
Here$\left( {{x_1},{y_1}} \right)$ is $\left( {1.5, - 13.5} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\left( {4, - 13} \right)$
So, we get

$$1.5 = \dfrac{{4 + p}}{2} \\\ 1.5 \times 2 = 4 + p \\\ 3 - 4 = p \\\$$

$p = - 1$
Let’s find $q$
We know that

$${y_1} = \dfrac{{{y_2} + q}}{2} \\\ \- 13.5 = \dfrac{{ - 13 + q}}{2} \\\ \- 13.5 \times 2 = - 13 + q \\\ \- 27 + 13 = q \\\$$

So, we get
$q = - 14$
So, the final answer is,
The mirror image (reflection) of point $\left( {4, - 13} \right)$ about the line $5x + y + 6 = 0$, will be$\left( { - 1, - 14} \right)$.
So, the correct answer is $C)$ $\left( { - 1, - 14} \right)$

So, the correct answer is “Option C”.

Note: Remember the formula with the involvement of two points with slope. When multiplying different sign terms we would remember the following things,
1. When a negative number is multiplied with a negative number the final answer will be a positive number.
2. When a positive number is multiplied with a positive number the final answer will be a positive number.
3. When a positive term is multiplied with the negative term the final answer will be a negative term.