Finding reflection of a point across a line - co ordinate geometry

What is the reflection of (a,b) across y = mx + c?
For e.g. 
Reflection of (a,b) across y = x + 1.

First draw a line passing through (a,b) perpendicular to y = x + 1.
That will have slope = -1

So, y = -x + k => b = -a + k => k = a + b
So 
y = -x + a + b is the line.

Now find intersection of these 2 lines.
=>
a + b - x = x + 1 => 2x = a + b - 1 => x = (a + b - 1)/2, y = a + b + (1 - a - b)/2 = (a + b + 1)/2
Now, 
this intersection point is the midpoint of (a,b) and its reflection (a',b').
So,
(a ' + a)/2 = (a + b - 1)/2 => a' = b - 1
(b ' + b)/2 = (a + b + 1)/2 => b' = a + 1

So,
reflection of (a,b) across y = x + 1 is (b-1,a+1).

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