Question

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

• A. a circle
• B. an ellipse
• C. a parabola
• D. a hyperbola

Hint:

In geometric problems involving tangency, the locus of the center of a circle touching a line and another fixed circle often leads to a parabolic path.

Answer 1

The Correct Option is C

To solve this problem, we will consider the given conditions one by one: 

- Condition 1: The circle touches the x-axis. When a circle touches the x-axis, the distance from its center to the x-axis is equal to its radius. So, if the center of the circle is \( (h, k) \), then the radius \( r \) of the circle is \( k \). This means that the y-coordinate of the center of the circle is equal to the radius, i.e., \( k = r \). 

- Condition 2: The circle touches another circle with center \( (0, 3) \) and radius 2. For two circles to touch each other externally, the distance between their centers is equal to the sum of their radii. Let the radius of the circle we are concerned with be \( r \) and its center be \( (h, k) \). The distance between the center of this circle \( (h, k) \) and the center of the given circle \( (0, 3) \) must be equal to the sum of their radii, i.e., \( r + 2 \). The distance between the two centers is calculated using the distance formula: \[ \text{Distance between centers} = \sqrt{(h - 0)^2 + (k - 3)^2} = r + 2 \] Simplifying this: \[ \sqrt{h^2 + (k - 3)^2} = r + 2 \] Squaring both sides: \[ h^2 + (k - 3)^2 = (r + 2)^2 \] Expanding: \[ h^2 + (k^2 - 6k + 9) = r^2 + 4r + 4 \] Now, considering that the circle also touches the x-axis (so \( k = r \)), we can substitute \( k = r \) into the equation: \[ h^2 + (r^2 - 6r + 9) = r^2 + 4r + 4 \] Simplifying: \[ h^2 - 6r + 9 = 4r + 4 \] \[ h^2 = 10r - 5 \] Thus, the locus of the center \( (h, k) \) of the circle satisfies the equation \( h^2 = 10r - 5 \). This is the equation of a parabola, which describes the path followed by the center of the circle. Hence, the locus of the center of the circle is a parabola, and the correct answer is \( \boxed{(c) \text{a parabola}} \).