First slide

Definition of a circle

Question

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

Moderate

A

$0 < k < 1 / 2$
B

$k \geq 1 / 2$
C

$- 1 / 2 \leq k \leq 1 / 2$
D

$k \leq 1 / 2$

Solution

Equation of the circle with centre $(h, k)$ and touching $x -$ axis is
$(x - h)^{2} + (y - k)^{2} = k^{2}$
which passes through $(- 1, 1)$
if $h^{2} + 2 h - 2 k + 2 = 0$
or if $(h + 1)^{2} = 2 k - 1$
For real values of $h, 2 k - 1 \geq 0$ .
$\Rightarrow k \geq 1 / 2$

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