JEE Main · 2007 · 29 Apr 2007 · Offline · Circle

Consider a family of circles which are passing through the point

(- 1, 1)

and are tangent to

x

axis. 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) $0 < k < 1/2$
(B) $k \geq \frac{1}{2}$ ✓ Correct
(C) $- 1/2 \leq k \leq 1/2$
(D) $k \leq 1/2$

Answer: (B)

Solution

A circle touching the

x

-axis with centre

(h, k)

has radius

∣ k ∣

, and since it passes through

(- 1, 1)

above the axis,

k > 0

and the radius is

k

. Substituting the point into

(x - h)^{2} + (y - k)^{2} = k^{2}

gives

h^{2} + 2 h + 2 - 2 k = 0

. For a real centre

h

the discriminant must satisfy

4 - 4 (2 - 2 k) \geq 0

, which reduces to

k \geq \frac{1}{2}

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option

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