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Question

The line x=y touches a circle at the point (1,1). If the circle also passes through the point (1,3), then its radius (in units) is :

A
2
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B
3
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C
22
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D
32
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Solution

The correct option is C 22
We know that, the equation of family of circles touching a line L at (x1,y1) is :
(xx1)2+(yy1)2+λL=0
(x1)2+(y1)2+λ(xy)=0

The circle also passes through (1,3).
0+16+4λ=0λ=4

Therefore, the equation of the circle is x2+y26x+2y+2=0r=(3)2+122=22

Alternate Solution:

Coordinates of the centre will be of form (b,1)
Now equation of normal through (1,1) is x+y=2
It will pass through the centre of the circle.
Hence b=3
Radius =22+22=22 units

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