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.

 Video Solution
Areas Related To Circles
Ex 12.1 | Question 2

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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