First slide

Equation of line in Straight lines

Question

Reflection of the point $(1, 1)$ with respect to the line $4 x + 3 y - 5 = 0$ is $(α, β)$ then $α + β$

Easy

A

$\frac{22}{15}$
B

$\frac{22}{25}$
C

$\frac{25}{22}$
D

$\frac{15}{22}$

Solution

If $(h, k)$ is the image of the point $(x_{1}, y_{1})$ with respect to the line $ax + by + c = 0$ then $\frac{h - x_{1}}{a} = \frac{k - y_{1}}{b} = - \frac{2 ({ax}_{1} + {by}_{1} + c)}{a^{2} + b^{2}}$
Hence, if $(α, β)$ is the image of the point $(1, 1)$ with respect to the line $4 x + 3 y - 5 = 0$ then $\frac{α - 1}{4} = \frac{β - 1}{3} = - \frac{2 (4 (1) + 3 (1) - 5)}{4^{2} + 3^{2}}$
Simplify $\frac{α - 1}{4} = \frac{β - 1}{3} = - \frac{4}{25}$

It implies
$\begin{matrix} \frac{α - 1}{4} = - \frac{4}{25} \\ α - 1 = - \frac{16}{25} \\ α = \frac{9}{25} \end{matrix}$
and
$\begin{matrix} \frac{β - 1}{3} = - \frac{4}{25} \\ β - 1 = - \frac{12}{25} \\ β = \frac{13}{25} \end{matrix}$
Therefore, $α + β = \frac{13}{25} + \frac{9}{25} = \frac{22}{25}$

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