If the area of a circle is equal to sum of the areas of two circles of diameters 10cm and 24 cm, then the diameter of the large circle $( in\ cm)$ is:

$( A) 34$
$( B) 26$
$( C) 17$
$( D) 14$

Given: Two circles of diameters $10\ cm$ and $24\ cm$. and the sum of the area of both circles is equal to the area of a third circle.

To do: To find out the diameter of the large$( third)$ circle.

Solution: As given in the question,

Diameters of the given two circles are $d_{1} =10\ cm$ and $d_{2} =24\ cm$

Let the radius of the both circles are $r_{1}$ and $r_{2}$ respectively,

$\Rightarrow r_{1} =\frac{d_{1}}{2} =\frac{10}{2} =5\ cm$ and $r_{2} =\frac{d_{2}}{2} =\frac{24}{2} =12\ cm$

Let us say, radius of the larger circle is $r$.

As we know area of a circle with radius $r,\ A=\pi r^{2}$

$\therefore \ A=\pi r^{2} =\pi ( r\ _{1})^{2} +\pi ( r_{2})^{2}$

$\Rightarrow \pi r^{2} =\pi ( 5)^{2} +\pi ( 12)^{2} =25 \pi +144\pi$

$\Rightarrow \pi r^{2} =169 \pi$

$\Rightarrow r^{2} =\frac{169 \pi}{\pi }$

$\Rightarrow r=\sqrt{169} =\pm 13$ 

Since radius can't be negative. 

$\therefore $ We reject $r=-13$ 

$\Rightarrow r=13\ cm$ and diameter of the larger circle$=2\times r=2\times 13=26\ cm$

$\therefore$ Option $( B)$ is correct.
 

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Updated on: 10-Oct-2022

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