Circle
Tangent Circles
IIT-JEE 2005 Screening
Grade 11
Question:
A circle is given by $x^2 + (y - 1)^2 = 1$. Another circle $C$ touches it externally and also the $x$-axis. The locus of the centre of $C$ is:
(1) $\{(x, y) : x^2 = 4y\} \cup \{(0, y) : y \leq 0\}$
(2) $\{(x, y) : y = x^2\}$
(3) $x^2 + (y - 1)^2 = 4$
(4) $\{(x, y) : x^2 = 4(y + 1)\}$
Step-by-Step Solution
Key Concept: Let the centre of $C$ be $(h, k)$ with radius $|k|$ (tangent to $x$-axis). External tangency gives $\sqrt{h^2 + (k - 1)^2} = |k| + 1$. Squaring: $h^2 = 4k$ for $k > 0$, or $h = 0$ with $k \leq 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)