Question

A ray of light passing through the point $(1,2)$ is reflected on the $x$-axis at a point $P$ and passes through the point $(5,3)$. The abscissa of the point $P$ is

• A. $3$
• B. $\frac{13}{3}$
• C. $\frac{13}{5}$
• D. $\frac{13}{4}$

Answer 1

The correct option is C $\frac{13}{5}$

The ray passes through $A(1,2)$ and reflects at $B$ and passes through $(5,3)$

The image of $(5,3)$ with respect to $x-axis$ is $(5,-3)$

The line $AB$ passes through the image of $(5,3)$

The slope of line is $\frac{2+3}{1-5}=\frac{-5}{4}$

So the equation of line is $y-2=\frac{-5}{4}(x-1)$

$4y-8=-5x+5$

$⟹5x+4y-13=0$

The line touches $x-axis$ at $y=0$

$⟹5x+4\left(0\right)-13=0$

$⟹5x=13$

$⟹x=\frac{13}{5}$