Question

Two circles touch internally. The sum of their areas is 116 sq. cm and the distance between their centres is 6 cm. Find the radii of the given circles.

Answer 1

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Answer 2

Let O and O1 be the centres of the circles.

Let x cm be the radius of the circle with centre O1. Then x+6cm is the radius of the circle with centre O.

Given, sum of the areas of these two circles = $\sf{116\pi \: {cm}^{2}}$

$\longrightarrow$ $\sf{\pi \: {x}^{2} + \pi {(x + 6})^{2} = 116\pi}$

$\longrightarrow$ $\sf{{x}^{2} + {(x + 6})^{2} = 116(dividing\:both\:sides\:by\:π)}$

$\longrightarrow$ $\sf{{x}^{2} + {x}^{2} + 12x + 36 = 116}$

$\longrightarrow$ $\sf{ {x}^{2} + 6x - 40 = 0}$

$\longrightarrow$ $\sf{(x - 4)(x + 10) = 0}$

As x ≠ -10,so x=4 and x+6=10

Hence, the required radii are 4cm and 10cm.