A line meets the coordinate axes in points A&B. A circle is circumscribed about the trangle OAB. If the perpendicular distances of the tangent drawn to the circle at origin 'O' from the points A & B are 6 & 2 units then the diameter of the circle is ____
 
 

Asked by Sunil Soni | 10th Aug, 2016, 05:13: PM

Expert Answer:

 
Let space straight Q space be space the space centre space of space the space circle space and space straight r space be space the space radius space of space the space circle.
Let space straight P comma space straight R space and space straight S space be space the space points space at space which space circle space touches space the space sides space OA comma space OB space and space AB space respectively.

Join space QP space and space QR.
As space QP space and space QR space are space radius space of space the space circle space so space they space are space perpendicular space to space the space tangents space OA space and space OB.
Also space it space is space given space that space OA space and space OB space are space perpendicular space to space each space other space at space straight O.

So space in space square OPQR space comma
angle straight O equals angle straight P equals angle straight R space equals space 90 degree
rightwards double arrow angle straight Q space equals space 90 degree
rightwards double arrow space square OPQR space space is space straight a space square.
rightwards double arrow OP space equals space OR space equals space straight r space

We space know space that space tangents space drawn space from space an space external space point space to space the space circle space touches space the space circle space at space same space distance.
therefore OR space equals space OP space equals space straight r
PA space equals SA space equals space 6 minus straight r
and space BS space equals space BR space equals space 2 minus straight r
So space AB space equals PA space plus space BR space equals space 6 minus straight r space plus space 2 space minus space straight r space equals 8 minus space 2 straight r.............. left parenthesis straight i right parenthesis

In triangle OAB comma space using space pythagoras space theorem space we space can space say comma
open parentheses AB close parentheses squared equals open parentheses OA close parentheses squared plus open parentheses OB close parentheses squared
open parentheses AB close parentheses squared equals space 36 plus space 4 space equals 40
AB space equals space 2 square root of 10............... left parenthesis ii right parenthesis

Using space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis space we space get
8 minus space 2 straight r space equals space space 2 square root of 10
2 straight r space space equals space 8 space minus space 2 space square root of 10 space units


therefore space The space diametre space of space the space circle space is space 8 minus 2 square root of 10 space space units.
 
 

Answered by Vijaykumar Wani | 11th Aug, 2016, 11:19: AM

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