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Q.
The equation of the line segment $A B$ is $y=x$. If $A$ and $B$ lie on the same side of the line mirror $2 x - y =1$, then the image of $AB$ has the equation
Straight Lines
Solution:
We have, to find the locus of the point $(h, k)$ whose image on the line $2 x - y -1=0$ lies on the line $y = x$.
Now, the image of $( h , k )$ on the line $2 x - y -1=0$ is given by
$\frac{ x _{2}- h }{2}=\frac{ y _{2}- k }{-1}=-\frac{2(2 h - k -1)}{5}$
or $x_{2}=\frac{-3 h+4 k+4}{5}$
and $y_{2}=\frac{4 h+3 k-2}{5}$
The point lies on $y = x$. Then,
$\frac{-3 h+4 k+4}{5}=\frac{4 h+3 k-2}{5}$
or $7 h - k =6$