Question

A circle passes through the origin O and cuts the axis at A(a, 0) and B(0, b). The reflection of O in the line AB is the point

• A. $[\frac{2a{b}^2}{a^2+b^2},\frac{2a^{2}b}{a^2+b^2}]$
  
• B. $[\frac{2a^{2}b}{a^2+b^2},\frac{2a{b}^2}{a^2+b^2}]$
  
• C. $[\frac{2ab}{a^2+b^2},\frac{2ab}{a^2+b^2}]$
  
• D. $(a,b)$
  

Answer 1

The correct option is A

$[\frac{2a{b}^2}{a^2+b^2},\frac{2a^{2}b}{a^2+b^2}]$

Equation of AB is x/a+y/b=1. If P(h,k) is the reflection of O in AB then (h/2, k/2) lies on AB and AB and OP are at right angles.
$\therefore \frac{h}{a}+\frac{k}{b}=2$
$-\frac{b}{a}\times \frac{k}{h}=-1$
$\Rightarrow h=\frac{2a{b}^2}{a^2+b^2}k=\frac{2a^{2}b}{a^2+b^2}$