Question

If arcs of same length in two circles, subtends angles of ${90}^{\circ }$ and ${135}^{\circ }$ at their centers, then the ratio of their radii is

• A. $2:1$
• B. $3:2$
• C. $4:3$

Answer 1

The correct option is B $3:2$

$\text{Let radii of circles be}{r}_1\&r_2\text{and lengths of arcs be}{l}_1,l_2\text{respectively.}$

Now, length of arc is given as:
$l=\frac{\theta }{{360}^{\circ }}\times 2\pi \times r$

For circle with angle subtended as $\theta ={90}^{\circ }$
$l_1=\frac{{90}^{\circ }}{{360}^{\circ }}\times 2\pi r_1$

Similarly, for circle with angle subtended as $\theta ={135}^{\circ }$
$l_2=\frac{{135}^{\circ }}{{360}^{\circ }}\times 2\pi r_2$

Now, since the length of arcs are same $\Rightarrow l_1=l_2$
$\Rightarrow \frac{{90}^{\circ }}{{360}^{\circ }}\times 2\pi r_1=\frac{{135}^{\circ }}{{360}^{\circ }}\times 2\pi r_2$

$\Rightarrow \frac{1}{4}\times r_1=\frac{3}{8}\times r_2$

$\Rightarrow \frac{r_1}{r_2}=\frac{3}{2}$

Hence, the ratio of their radii $=3:2$