First slide

Equation of line in Straight lines

Question

A ray of light is sent along the line which passes through the point (2, 3). The ray is reflected from the point P on x-axis. If the reflected ray passes through the point (6, 4), then the coordinates of P are

Moderate

A

$(\frac{26}{7}, 0)$
B

$(0, \frac{26}{7})$
C

$(- \frac{26}{7}, 0)$
D

none of these

Solution

Let $P \equiv (α, 0)$ .
Let the reflected ray makes an angle $θ$ with +ve direction of x-axis, then the incident ray makes angle $(π - θ)$ with positive direction of x-axis.
The slope of the incident ray is
$= \frac{0 - 3}{α - 2} = \tan (π - θ)$ i.e., $\tan θ = \frac{3}{α - 2}$
The slope of the reflected ray is
$= \frac{4 - 0}{6 - α} = \tan θ$ i.e., $\tan θ = \frac{4}{6 - α}$
From Eq. (1) and (2), we get
$\frac{3}{α - 2} = \frac{4}{6 - α} \Rightarrow 18 - 3 α = 4 α - 8$
$\Rightarrow 7 α = 26 o r α = \frac{26}{7}$
$∴$ The coordiantes of A are $(\frac{26}{7}, 0)$

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