Question

Consider a family of circles which are passing through the point (–1, 1) and are tangent to x-axis. (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval

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

Answer 1

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

Since, circles are passing through the point (–1, 1) and touching X-axis, the equationof circles can be written as -
$(h+1{)}^2+(K–1{)}^2=K^2$
$h^2+2h+2–2K=0$
If h is real the $\Delta \ge 0$
4- 4.1. (2 – 2K) $\ge 0$
= 4 - 8 + 8k $\ge 0$
= 8k $\ge 4$
= $k\ge \frac{1}{2}$