First slide

Definition of a circle

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:

NA

A

$k \geq \frac{1}{2}$
B

$- \frac{1}{2} \leq k \leq \frac{1}{2}$
C

$k \leq \frac{1}{2}$
D

$0 < k < \frac{1}{2}$

Solution

$Equation of family of circles with centre (h, k) and touching x -axis is:$
${(x - h)}^{2} + {(y - k)}^{2} = k^{2}$
$\Rightarrow x^{2} + y^{2} - 2 h x - 2 k y + h^{2} = 0$
$As M (1, 1) lies on it$
$\Rightarrow 2 - 2 h - 2 k + h^{2} = 0$
$\Rightarrow k = \frac{{(h - 1)}^{2} + 1}{2} \geq \frac{1}{2}$
$∴ k \geq \frac{1}{2}$

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