| Assertion (A): If the circumferences of two circles are in the ratio \[\text{1}:\text{3},\] then the ratio of their areas is \[1:9\]. |
| Reason (R): The area of a sector of a circle of radius r with sector angle \[\theta \] is \[\frac{\theta }{360{}^\circ }\times \pi {{r}^{2}}\]. |
A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: A
Solution :
| [a] Let \[{{r}_{1}}\] and \[{{r}_{2}}\] be the radii of two circles. |
| Then, ratio of their circumference \[=\frac{2\pi {{r}_{1}}}{2\pi {{r}_{2}}}=\frac{1}{3}\] |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{1}{3}\,\,\,\,\Rightarrow \,\,\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{1}{9}\] |
| Now, ratio of their areas |
| \[=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}=\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{1}{9}=1:9\]. |
| \[\therefore \] Assertion: True; Reason: True but it is not the correct explanation of Reason. |
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