Q. Consider a family of circles, which are passing through the point $( - 1,\ 1)$ and are tangent to $x$-axis. If $(h,\ k)$ are the coordinates of the centre of the circles, then the set of values of $k$ is given by the interval
A) $0 < k < \frac{1}{2}$
B) $k \geq \frac{1}{2}$
C) $- \frac{1}{2} \leq k \leq \frac{1}{2}$
D) $k \leq \frac{1}{2}$
Explanation
(b)
Equation of circle which touches $x$-axis and coordinates of centre are $(h,\ k)$ is
$(x - h)^{2} + (y - k)^{2} = k^{2}$
Since, it is passing through $( - 1,\ 1)$, then
$( - 1 - h)^{2} + (1 - k)^{2} = k^{2}$
$\Rightarrow \ \ h^{2} + 2h - 2k + 2 = 0$
For real circles, $D \geq 0$,
$\Rightarrow \ \ \ \ (2)^{2} - 4( - 2k + 2) \geq 0\ \Rightarrow \ \ k \geq \frac{1}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.