Question

Consider a family of circles which are passing through the point (–1, 1) and has x-axis as its tangents. If (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. 0 $<$ k $<$ $\frac{1}{2}$
• B. k $\ge$ $\frac{1}{2}$
• C. $-\frac{1}{2}$$\le$ k $\le$ $\frac{1}{2}$
• D. k $\le$ $\frac{1}{2}$

Answer 1

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


We have r = CP $\Rightarrow$ $\left|k\right|$ = CP
$\Rightarrow k^2=(h+1{)}^2+(k-1{)}^2$
$\Rightarrow h^2$ + 2h + 2(2 - 2k) = 0 $\to$ (1)
which is a quadratic equation in `h'
Since h is real.
4 $\ge$ 4(1) (2 - 2k) $\Rightarrow$ $\frac{1}{2}$ $\ge$ 1 - k $\Rightarrow$ - $\frac{1}{2}$ $\ge$ - k