Question

Consider a family of circles which are passing through $M(1,1)$ and are tangent to x-axis. If $(h,k)$ is the centre of circle, then

• A. $k\ge \frac{1}{2}$
• B. $-\frac{1}{2}\le k\le \frac{1}{2}$
• C. $k\le \frac{1}{2}$
• D. $0<k<\frac{1}{2}$

Answer 1

The correct option is A $k\ge \frac{1}{2}$

Equation of circle with centre $(h,k)$ and to using x-axis is given by

$(x-h{)}^2+(y-k{)}^2=k^2$

$\Rightarrow x^2+h^2-\left(2h\right)x+y^2+k^2-\left(2k\right)y=k^2$

$\Rightarrow x^2+y^2-\left(2h\right)x-\left(2k\right)y+h^2=0$

It passes through the point $(1,1)$

$\Rightarrow 1^2+1^2-2h-2k+h^2=0$

$\Rightarrow h^2-2h+(2-2k)=0$

As value of h is real, $D\ge 0$

$\Rightarrow (-2{)}^2-4(2-2k)\ge 0$

$\Rightarrow k\ge \frac{1}{2}$.