JEE Main · 2016 · 03 Apr 2016 · Offline · Circle

The centres of those circles which touch the circle,

x^{2} + y^{2} - 8 x - 8 y - 4 = 0

, externally and also touch the x-axis, lie on:

(A)an ellipse which is not a circle.
(B)a hyperbola.
(C)a parabola.✓ Correct
(D)a circle.

Answer: (C)

Solution

Let a circle of centre

(h, k)

touch the

x

-axis, so its radius is

∣ k ∣

. The given circle,

(x - 4)^{2} + (y - 4)^{2} = 36

, has centre

(4, 4)

and radius 6, and external contact means the centre-to-centre distance equals the sum of radii:

(h - 4)^{2} + (k - 4)^{2} = ∣ k ∣ + 6

. Squaring and expanding, the

k^{2}

terms cancel, leaving a relation linear in

k

but quadratic in

h

, for example

h^{2} - 8 h - 20 k - 4 = 0

when

k \geq 0

. A locus where the square of one coordinate is tied to the first power of the other is a parabola.

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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