First slide

Equation of line in Straight lines

Question

A ray of light passing through the point $(3, 7)$ reflects on the $x -$ axis at a point A and the reflected ray passes through the point $(2, 5)$ , the coordinates of $A$ are

Moderate

A

$(\frac{29}{12}, 0)$
B

$(\frac{1}{2}, 0)$
C

$(- \frac{1}{2}, 0)$
D

$(- \frac{29}{12}, 0)$

Solution

Let $p$ be the given point $(3, 7)$ and the coordinates of $A$ be $(x, 0)$ so, if the ray $P A$ makes an angle $θ$ with the positive direction of $x -$ axis, the reflected ray $A Q$ , where $Q$ is $(2, 5)$ , will make the same angle $q$ with the negative direction of $x -$ axis. So the slope of $P A = -$ (the slope of $Q A$ )
$\begin{array}{l} \Rightarrow \frac{7}{3 - x} = - \frac{5}{2 - x} \\ \Rightarrow 7 (2 - x) + 5 (3 - x) = 0 \\ \Rightarrow x = \frac{29}{12} \end{array}$
$\Rightarrow$ coordinates of $A$ are $(\frac{29}{12}, 0) .$

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