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Question

The circle x2+y24x4y+4=0 is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is x+yxy+kx2+y2=0, then the value of k is equal to

A
2
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B
1
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C
3
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D
2
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Solution

The correct option is B 1
Let OAB be the triangle in which the circle x2+y24x4y+4=0 is inscribed.
Let points A=(a,0) and B=(0,b)
So, the equation of the line AB is
bx+ayab=0


Distance from the centre (2,2) to the line AB is equal to radius,
|2a+2bab|a2+b2=2|2(a+b)ab|=2a2+b2

As origin and C(2,2) lies on the same side of line AB, so
2(a+b)ab<0
So,
2(a+b)+ab=2a2+b22a+2bab+2a2+b2=0(1)

Let P(h,k) be the circumcentre of OAB
As OAB is a right angled triangle at O, so circumcentre is the mid point of line segment AB
h=a2, k=b2a=2h, b=2k
Putting in equation (1), we get
4h+4k4hk+4h2+k2=0
Locus of the circumcentre is
x+yxy+x2+y2=0

Hence, k=1

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