Circles
Circle
Allen Star Batch
Grade 11
Question:
If the $C_1, C_2, C_3$ and $C$ are four circles of radius $r_1, r_2, r_3, r$ respectively as shown in figure:
radius of the circle $C$ is $2\sqrt{r_1r_3}$
$\tan\frac{\theta}{2} = \frac{r_3 - r_1}{2\sqrt{r_1r_2}}$
$PQ = \frac{2r_1^{3/2}r_2^{1/2}}{r_2 - r_1}$
If $r_1 = 2$ and $r_1 + r_2 + r_3 = 14$ then $\frac{r_1}{r_2} = 2$
Step-by-Step Solution
Key Concept: Chord lengths in circles are computed directly from radii using the perpendicular distance from the center to the chord.
Calculate the distances from each point to the center using the distance formula. For points like $Q$, $R$, $C$, $S$, and $QS$ on the given configuration, each chord length equals $2\sqrt{r_i^2}$ where $r_i$ is the radius of the respective circle. The calculations yield $QR = 2\sqrt{r_1^2}$, $RC = 2\sqrt{r_2^2}$, $CS = 2\sqrt{r_3^2}$, $RS = 2\sqrt{r_4^2}$, $QS = 2\sqrt{r_5^2}$, and $QE = 2\sqrt{r_6^2}$ for their respective circles.
Correct Answer: 2,3