Question

The difference of areas of two sectors of different circles is $77{\text{cm}}^2$. If the radii of the circles are $21\text{cm}$ and $14\text{cm}$ respectively, and corresponding angles of the sectors made at the centre are ${60}^{\circ }$and $\theta$, then the value of $\theta$ can be

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${90}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is C ${90}^{\circ }$

Let $r_1$ and $r_2$ be the radii of two circles.
$\therefore r_1=21\text{cm},r_2=14\text{cm}$

Since, the area of a sector making an angle $\alpha$ at the centre of a circle of radius $r$ is given as $\frac{\alpha }{{360}^{\circ }}\times \pi r^2$

Given that the difference of areas of two sectors of different circles is $77{\text{cm}}^2$.
$\Rightarrow \frac{{60}^{\circ }}{{360}^{\circ }}\times \pi r_1^2-\frac{\theta }{{360}^{\circ }}\times \pi r_2^2=77$

$\Rightarrow \frac{{60}^{\circ }}{{360}^{\circ }}\times \pi (21{)}^2-\frac{\theta }{{360}^{\circ }}\times \pi (14{)}^2=77$
$\Rightarrow \frac{1}{6}\times \frac{22}{7}\times (21{)}^2-\frac{\theta }{{360}^{\circ }}\times \frac{22}{7}\times (14{)}^2=77$
$\Rightarrow 11\times 21-11\times 7\times \frac{\theta }{{45}^{\circ }}=77$
$\Rightarrow 11(21-\frac{7\theta }{{45}^{\circ }})=11\times 7$
$\Rightarrow 21-\frac{7\theta }{{45}^{\circ }}=7$

$\Rightarrow \frac{7\theta }{{45}^{\circ }}=21-7$

$\Rightarrow \frac{7\theta }{{45}^{\circ }}=14$

$\Rightarrow \theta =\frac{14}{7}\times {45}^{\circ }={90}^{\circ }$


Hence, the correct answer is option c.