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Question

A line meets the coordinate axes in A and B. A circle is circumscribed about the AOB. If m,n are the distance of the tangent to the circle at the origin from the points A and B, respectively the diameter of the circle is

A
m(m+n)
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B
m+n
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C
n(m+n)
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D
None of these
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Solution

The correct option is A m+n
Let A(a,0) and B(0,b)
Since AOB=900, AB is the diameter
Center of the circle is and radius =12a2+b2.
Equation of the circle is
(xa2)2+(yb2)2=14(a2+b2)x2+y2axby=0
Equation of tangent to the circle at O(0,0) is ax+by=0 ...(1)
m= length of from A(a,0) on (1) =a2a2+b2
and n= length from (0,b) on (1) =b2a2+b2
Diameter a2+b2=m+n

389325_257316_ans_2549948c6d8d402dba0d146592263bed.png

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