A circle touches the x -axis and also touches the circle which centre at (0,3) and radius 2 . The locus of the centre of the circle is

Question

A circle touches the x -axis and also touches the circle which centre at (0,3) and radius 2 . The locus of the centre of the circle is (A) a parabola (B) a circle (C) an ellipse (D) a hyperbola

Tardigrade Admin · Accepted Answer

Let the centre of a circle be C1(h, k) Since, C1 C2=r1 +r2 (given) ⇒ √(h-0)2+(k-3)2=|k+2| ⇒ h2=5(2 k-1) Hence, locus of a point is x2=5(2 y-1) which represents a equation of parabola.

Q. A circle touches the $x$ -axis and also touches the circle which centre at $(0, 3)$ and radius $2.$ The locus of the centre of the circle is

a parabola

a circle

an ellipse

a hyperbola

Let the centre of a circle be $C_{1} (h, k)$
Since, $C_{1} C_{2} = r_{1} + r_{2}$ (given)
$\Rightarrow (h - 0)^{2} + (k - 3)^{2} = ∣ k + 2∣$
$\Rightarrow h^{2} = 5 (2 k - 1)$
Hence, locus of a point is $x^{2} = 5 (2 y - 1)$ which represents a equation of parabola.

Q. A circle touches the x -axis and also touches the circle which centre at (0,3) and radius 2. The locus of the centre of the circle is