 Hello and welcome to the session. In this session we discussed the following question which says two circles touch internally. The sum of their areas is 125 pi cm2 and the distance between their centers is 5 cm. Find the radii of the circles. So we are given that these two circles touch internally and sum of their areas is also given to us. We are also given the distance between their centers and we have to find the radii of these circles. So first of all let's recall the formula for the area of a circle and this is equal to pi r square where this r is the radius of the circle. This is the key idea to be used in this question. Now let's proceed with the solution. Now as you can see these two circles are touching each other internally at a point A. We have one circle with center as O and the other circle that is the internal circle has the center as O dash and we are given that the distance O O dash that is the distance between the centers of the two circles is 5 cm. So this means that O O dash is equal to 5 cm. Now let the radius of the circle with center as O that is the outer circle be equal to r1 and suppose the radius of the circle with center as O dash that is this internal circle be equal to r2 minus the distance between the two centers of the circle that is O O dash is equal to 5 cm. So this would obviously mean that the difference between the radii of the two circles that is r1 minus r2 is equal to 5 cm and this would mean that r1 is equal to r2 plus 5 cm. So if we have that the circle with center as O dash has radius r2 then the radius of the circle with center as O is equal to r1 that is equal to r2 plus 5 cm. So the internal circle that is which has center as O dash has radius r2 and the outer circle with center O has radius r2 plus 5. Now let's calculate the areas of both the circles. So area of the circle with center as O is equal to pi r1 square that is equal to pi r2 plus 5 whole square cm square and the area of the circle with center as O dash is equal to pi r2 square cm square. Now it's given that the sum of the areas of two given circles is equal to 125 pi cm square. This means that pi r2 square plus pi into r2 plus 5 whole square is equal to 125 pi. Further we get pi common here r2 square plus r2 plus 5 whole square is equal to 125 pi. This pi cancels with this pi. So we get r2 square plus r2 square plus 10 r2 plus 25 is equal to 125. This gives us 2 r2 square plus 10 r2 minus 100 is equal to 0 that is we have r2 square plus 5 r2 minus 50 is equal to 0. So now splitting the middle term of this quadratic equation we get r2 square minus 5 r2 plus 10 r2 minus 50 is equal to 0. Now r2 common from the first group so inside we have r2 minus 5 plus 10 common from the second group inside the bracket we have r2 minus 5 is equal to 0. This gives us r2 minus 5 this whole into r2 plus 10 is equal to 0. So this means r2 minus 5 equal to 0 or r2 plus 10 equal to 0 that is r2 equal to 5 or r2 equal to minus 10. Now as r2 is the radius and we know that radius cannot be negative therefore r2 is not equal to minus 10 and thus we have r2 is equal to 5. Thus we say the radius of the circle with center as o dash is equal to 5 centimeters that is r2 and the radius of the circle with center as o is equal to r2 plus 5 centimeters. So this would be equal to 5 plus 5 centimeters that is equal to 10 centimeters. So area of this internal circle is 5 centimeters and the center circle is 10 centimeters. So this is our final answer this completes the session. Hope you have understood the solution of this question.