Question

If the image of the point (2,1) with respect to the line mirror be (5, 2). Find the equation of the mirror.

Answer 1

Let CD be the line mirror with slope m_1.

Now the slope of the line joining A(2, 1) and B(5, 2).
m₂ = ${\displaystyle \frac{2-1}{5-2}=\frac{1}{3}}$
Since CD ⊥ AB
So, m₁m₂ = -1
⇒ ${\displaystyle m_1\times \frac{1}{3}}$ = -1
⇒ m₁ = -3.
Now mid point of AB = ${\displaystyle (\frac{2+5}{2},\frac{1+2}{2})=(\frac{7}{2},\frac{3}{2})}$
Equation of the mirror CD,
y = y₁ = m(x - x1)
⇒ ${\displaystyle y-\frac{3}{2}=-3(x-\frac{7}{2})}$
⇒ ${\displaystyle y-\frac{3}{2}=-3x+\frac{21}{2}}$
⇒ 2y - 3 = -6x + 21
⇒ 6x + 2y - 3 - 21 = 0
⇒ 6x + 2y - 24 = 0
or 3x + y - 12 = 0.