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 $α + β$

Moderate

A

$\frac{22}{15}$
B

$\frac{22}{25}$
C

$\frac{25}{22}$
D

$\frac{15}{22}$

Solution

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