Question

The reflection of the point $\left(4,-13\right)$ in the line $5x+y+6=0$ is

• A. $\left(-1,-14\right)$
• B. $\left(3,4\right)$
• C. $\left(1,2\right)$
• D. $\left(-4,13\right)$

Answer 1

The correct option is A

$\left(-1,-14\right)$

Explanation for the correct answer:

Compute the required value:

Let $A\left(4,-13\right)=\left(x_1,y_1\right)$

The reflection of a point $\left(x_1,y_1\right)$ about a line $ax+by+c=0$ is given by

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{2\left(a{x}_1+b{y}_1+c\right)}{a^2+b^2}$

Comparing $5x+y+6=0$ with $ax+by+c=0$ we get

$a=5,b=1,c=6$

Substituting all the given values

$\frac{x-4}{5}=\frac{y+13}{1}=-\frac{2\left(5\times 4+1\times \left(-13\right)+6\right)}{25+1}$

$\Rightarrow$$\frac{x-4}{5}=\frac{y+13}{1}=-1$

$\Rightarrow$ $\frac{x-4}{5}=-1$ and $\frac{y+13}{1}=-1$

$\Rightarrow$ $x-4=-5$ and $y+13=-1$

$\Rightarrow$ $x=-1$ and $y=-14$

Hence the reflection of the point $\left(4,-13\right)$ about the line $5x+y+6=0$ is $\left(-1,-14\right)$.

Hence, option $\left(A\right)$ is the correct answer.