Question

Equation of the circle which of the mirror image of the circle $x^2+y^2-2x=0$ in the line $x+y=2$ is

• A. $x^2+y^2-2x+4y+3=0$
• B. $2(x^2+y^2)+x+y+1=0$
• C. $x^2+y^2-4x-2y+4=0$
• D. None of the above

Answer 1

Answer: (C)

Given equation of circle and line are $x^2+y^2-2x=0$ .. (i) and $x+y=2$ ..(ii) Centre and radius of the circle are $(1,0)$ and1. Let the centre of the image circle be $(x_1{y}_1)$ . Hence, $(x_1{y}_1)$ be the image of the point $(1,0)$ with respect to the line

$=x+y=2,$ then $\frac{x_1-1}{1}=\frac{y_1-0}{1}=\frac{-2[1(1)+1(0)-2]}{{(1)}^2+{(1)}^2}$

$\Rightarrow$ $\frac{x_1-1}{1}=\frac{y_1}{1}=\frac{2}{2}=1$

$\Rightarrow$ $x_1=2,y_1=1$

$\therefore$ Equation of the imaged circle is ${(x-2)}^2+{(y-1)}^2=1^2$

$\Rightarrow$ $x^2+4-4x+y^2+1-2y=1$

$\Rightarrow$ $x^2+y^2-4x-2y+4=0$