two circles have a common external tangent of length 15 and a common internal tangent of length 7 the distance - Maths - Conic Sections

Question

two circles have a common external tangent of length 15 and a
common internal tangent of length 7 the distance between the
centres is 25 if the difference in the area of the two circles is
60k pi then the value of k is

Neha Sethi · Accepted Answer

Dear student

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 ∴∠O'CO=90° and OC=a-b ∵O'C=15=AB and ∠CAB=90°and d=25 units [Given[In △OCO' by pythagoras theorem, we haveO'C2+OC2=OO'2⇒a-b2+152=252⇒(a-b)2=625-225⇒(a-b)2=400 (1)Again, the length of common internal tangent to these circles is 7 units We can draw it as:

Here PQ = 7 units is the length of common internal tangent Draw a line O'R parallel to PQ so that it makes a right angled triangle OO'R with OR = OP + PR = a + b units O'R = PQ = 7 units and OO' = d units By Pythagoras theorem in △OO'ROO'2=(OR)2+(O'R)2⇒252=(a+b)2+72⇒a+b2=576 (2)Solving (1) and (2), we get

Since radius cannot be negative So, we take values of a and b as22,2,42,22,2,22←rejected ∵a>bCase 1: When a=42 and b=22Then Difference between areas of the two circles=Area of circle with radius is 42 units-Area of circle with radius is 22 units60kπ=π422-π222 ∵Area of circle=πr260k=1764-48460k=1280k=643Case 1: When a=22 and b=2Then Difference between areas of the two circles=Area of circle with radius is 22 units-Area of circle with radius is 2 units60kπ=π222-π22 ∵Area of circle=πr260k=484-460k=480k=8 Regards

two circles have a common external tangent of length 15 and a common internal tangent of length 7 the distance between the centres is 25 if the difference in the area of the two circles is 60k pi then the value of k is