AIEEE 2005 (Offline)Question 1
A circle touches the $x$-axis and also touches the circle with centre at $(0,3)$ and radius $2$. The locus of the centre of the circle is :
Circle
Solution
Concept. Write both touching conditions and eliminate the radius. Formulas used. (1) Touching the $x$-axis: radius $=k$ (take $k>0$).(2) External contact: distance between centres $=r_1+r_2$.Step 1 - Use (1), (2). $$\sqrt{h^2+(k-3)^2}=k+2.$$ Step 2 - Square. $$h^2+k^2-6k+9=k^2+4k+4\Rightarrow h^2=10k-5.$$ Step 3 - Identify. $x^2=10\left(y-\frac12\right)$: a parabola. Answer: D