Question

$S(x,y)=0$ represent a circle. The equation $S(x,2)=0$ gives two identical solutions $x=1$ and the equation $S(1,y)=0$ gives two distinct solutions $y=0,2$. the equation of the circle is

• A. $x^2+y^2+2x+2y+1=0$
• B. $x^2+y^2+2x+2y-1=0$
• C. $x^2+y^2-2x-2y+1=0$
• D. None of these

Answer 1

Answer: (C)

$x^2+y^2-2x-2y+1=0$

We have

$S(x,2)=0$ gives two identical solutions $x=1$

$\Rightarrow$ line $y=2$ is a tangent to the circle $S(x,y)=0$ at the point $(1,2)$

and, $S(1,y)=0$ gives two distinct solutions $y=0,2$

$\Rightarrow$ line $x=1$ cuts the circle $S(x,y)=0$ at the points, $(1,0)$ and $\left(1.2\right)$

Clearly, from the fig., the points $A(1,2)$ and $B(1,0)$ are diametrically opposite points.

Thus, equation of the circle, is

${(x-1)}^2+y(y-2)=0$

i.e., $x^2+y^2-2x-2y+1=0.$