Question

find the image of the point (2 1) with respect to the line mirror (1,4) having slope -1​

Answer 1

The line mirror passing through (1, 4) and having slope -1 should have the equation,

$\longrightarrow y-4=-1(x-1)$

$\longrightarrow x+y-5=0$

The point (2, 1) and its image wrt the line mirror, should be equidistant from the line mirror.

Let the image be (h, k). So we have (recall the equation of the distance of a point from a line),

$\longrightarrow\dfrac{|h+k-5|}{\sqrt{1^2+1^2}}=\dfrac{|2+1-5|}{\sqrt{1^2+1^2}}$

$\longrightarrow |h+k-5|=2$

Take the modulus in such a way that $h+k\neq 3$ (else we get (h, k) = (2, 1)). So,

$\longrightarrow h+k-5=2$

$\longrightarrow h+k=7\quad\quad\dots(1)$

The line joining (2, 1) and (h, k) should be perpendicular to the line mirror, so the slope of the line formed will be 1 (negative reciprocal).

$\longrightarrow k-1 =h-2$

$\longrightarrow h-k=1\quad\quad\dots(2)$

Solving (1) and (2) we get,

$\longrightarrow\underline{\underline{(h,\ k)=(4,\ 3 )}}$