Ex.12.1 Q2 Areas Related to Circles Solution - NCERT Maths Class 10

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Question

The radii of two circles are \(8\,\rm{cm}\) and \(6\,\rm{cm}\) respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Text Solution

What is known?

Radii of two circles.

What is unknown?

Radius of \(3^\rm{rd}\) circle.

Reasoning:

Using the formula of area of circle \(A = \pi {r^2}\) we find the radius of the circle.

Steps:

Radius of \((r_1)\) \(1^\rm{st}\) circle \(= 8\,\rm{cm}\)

Radius of \((r_2)\)\(2^\rm{nd}\) circle \(= 6\,\rm{cm}\)

Let the radius of \(3^\rm{rd}\) circle \(=r.\)

Area of \(1^\rm{st} \)circle \(=\pi \rm{r}_{1}^{2}= \pi (8)^2= 64\pi\)

Area of \(2^\rm{nd}\) circle \(=\pi \rm{r}_{2}^{2}= \pi (6)^2= 36\pi\)

Given that,

Area of \(3^\rm{rd}\) circle \(=\) Area of \(1^\rm{st}\)  circle \(+\) Area of \(2^\rm{nd}\) circle

\[\begin{align}{\pi{{{r}}^{{2}}}}& = {\pi {{r}}_{{1}}^{{2}}\,{{ + }}\,\pi {{r}}_{{2}}^{{2}}}\\{\pi {{{r}}^{{2}}}} &= {64\pi \,{{ + }}\,36\pi }\\{\pi{{{r}}^{{2}}}} &= {{100\pi }}\\{\,\,{{{r}}^{{2}}}} &={{{100}}}\\{\,\,{{r}}} &= \,\pm \,10\end{align}\]

However, the radius cannot be negative. Therefore, the radius of the circle having area equal to the sum of the areas of the two circles is \(10\,\rm{cm.}\)

  
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