KCET · Maths · Circle
If in two circles, arcs of the same length subtend angles \(30^{\circ}\) and \(78^{\circ}\) at the centre, then the ratio of their radii is
- A \(\frac{5}{13}\)
- B \(\frac{13}{5}\)
- C \(\frac{13}{4}\)
- D \(\frac{4}{13}\)
Answer & Solution
Correct Answer
(B) \(\frac{13}{5}\)
Step-by-step Solution
Detailed explanation
Let the radii of the two circles be \(r_1\) and \(r_2\). Let an arc of length \(l\) subtend an angle of \(30^{\circ}\) at the centre of the circle of radius \(r_1\), while let an arc of length \(l\) subtend an angle of \(78^{\circ}\) at the centre of the circle of radius \(r_2\).
Now, \(30^{\circ}=\frac{\pi}{6}\) radian and \(78^{\circ}=\frac{13 \pi}{30}\) radian
We know that in a circle of radius \(r\) unit, if an arc of length \(l\) unit subtend an angle \(\theta\) radian at the centre, then
\(\theta=\frac{l}{r}\) or \(l=r \theta\)
\(\therefore \quad l=\frac{r_1 \pi}{6}\) and \(l=\frac{r_2 13 \pi}{30}\)
\(\Rightarrow \quad \frac{r_1 \pi \pi}{6}=\frac{r_2 13 \pi}{30}\)
\(\Rightarrow \quad r_1=r_2 \frac{13}{5}\)
\(\Rightarrow \quad \frac{r_1}{r_2}=\frac{13}{5}\)
Now, \(30^{\circ}=\frac{\pi}{6}\) radian and \(78^{\circ}=\frac{13 \pi}{30}\) radian
We know that in a circle of radius \(r\) unit, if an arc of length \(l\) unit subtend an angle \(\theta\) radian at the centre, then
\(\theta=\frac{l}{r}\) or \(l=r \theta\)
\(\therefore \quad l=\frac{r_1 \pi}{6}\) and \(l=\frac{r_2 13 \pi}{30}\)
\(\Rightarrow \quad \frac{r_1 \pi \pi}{6}=\frac{r_2 13 \pi}{30}\)
\(\Rightarrow \quad r_1=r_2 \frac{13}{5}\)
\(\Rightarrow \quad \frac{r_1}{r_2}=\frac{13}{5}\)
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