Circles
Circle
Allen Star Batch
Grade 11
Question:
Two Circles of radii 36 units and 9 units touch each other externally, a third circle of radius $r$ touches the two given circles externally and also their common tangent, then the value of $r$ is:
4
5
$\sqrt{17}$
$\sqrt{18}$
Step-by-Step Solution
Key Concept: The sum of square roots of radii relates to the direct common tangent length formula.
Given $2\sqrt{r_1} + 2\sqrt{r_2} = 2\sqrt{rr_r}$, we simplify to get $\sqrt{r}(6+3) = 6 \times 3$. Solving yields $r = 4$. The length of the direct common tangent between two circles with radii $r_1$ and $r_2$ separated by distance $d$ is $\sqrt{d^2 - (r_1 - r_2)^2}$.
Correct Answer: 1