Question

Two circles touch internally. The sum of their areas is (116π) cm² and the distance between their centres is 6 cm. find the radii of the circles.

Answer 1

Heya dude here is ur answer....

Let a circle with center O
And radius R.
let
another circle inside the first circle  with center o' and radius r .
A/Q,Area of 1st circle + area of 2nd circle = 116π cm²
⇒ πR² + πr² = 116π 
⇒ π(R² + r²)  = 116π
⇒ R² + r² =116 ----------------(i)

Now,
Distance between the centers of circles = 6 cm

i.e, R - r = 6
⇒  R = r + 6  -------------------(ii)

From Eqn (i) & (ii),

(r + 6)² + r² = 116
⇒ r² + 12r +36 + r² =116
⇒ 2r² +12r +36 -116 = 0
⇒ 2r² +12r - 80 = 0
⇒ r² +6r - 40 = 0
⇒ r² +10r - 4r - 40 = 0
⇒ r(r + 10) - 4(r + 10) = 0
⇒ (r + 10)(r - 4) = 0

 hence  r = 4 cmr ≠ -10 cm  

     {∵ length can't be -ve}

Therefore radii of the circles are r = 4 cm

,R = 4 + 6 = 10 cm.

Tysm#kundan

Answer 2

The circles touch internally.


Therefore,


Difference of their radii = Distance between their centres = 6 cm.

Let the radii of the given circles be r cm and ( r + 6 ) cm.


Sum of their areas = 116 π



[ πr² + π( r + 6 )² = 116π



=> π [ r² + ( r + 6 )² = 116π




=> r² + ( r + 6 )² = 116


=> 2r² + 12r - 80 = 0



=> r² + 6r - 40 = 0



=> r² + 10r - 4r - 40 = 0


=> ( r + 10 ) ( r - 4 ) = 0


=> ( r + 10 ) = 0 or ( r - 4 ) = 0



=> r = -10 or r = 4

=> r = 4 [ Neglecting r = -10 , as radius can't be negative ]


Therefore,

The radii of the given circles are 4 cm and 10 cm.