Question

If the length of a common internal tangent to two circles is 7 and that of a common external tangent is 11 then the product of the radii of the two circles is

Answer 1

Answer:

The product of the radius of two circles is 18

Step-by-step explanation:

Please refer to the diagrams given in the attachment.

Let d be the distance of the centres of the circle. Let a be the radius of large circle and b be the radius of smaller circle.

From O', draw a line parallel to AB which meets OA and C.

In Right angled ΔOCO', we have –

O'C² + OC² = OO'²

d² = (a-b)²+ 11²

11² = d² - (a-b)²

121 = d² - a² - b² + 2ab -----> 1

Again, the length of common internal tangent to these two circles is 7 units. We can draw it as –

By Pythagoras theorem,

7² + (a+b)² = d²

49 = d² - a² - b² - 2ab ------> 2

Subtracting 2 from 1,

we get 121 - 49 = 4ab

ab = 18

Hence the product of radius of two circles is 18 square units.