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Question

Find the radius of smallest circle touching 3xy=6 at point (1,3)and also touches the line y=x ?

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Solution

Since we know that 3xy=6 touches the circle at (1,3) we can form the
structure of the equation of circle.
consider (x1)2+(y+3)2=0. It just satisfies one point (1,3) Add to it a
multiple of equation of the line.
(x1)2+(y+3)22λ(3xy6)=0. it is a circle of variable radius
which touches the given line at (1,3).
Expanding
x2+y22(1+3λ)+2(3+λ)+10+12λ=0--(1)
It should also touch y=x one can use the concept of perpendicular distance
from center equals radius. But intersection concept is much easier because of
simple equation y=x
By substituting y=xin(1)
x2+2(1λ)x+5+6λ=0
This must be a perfect since the tangent intersects circle at two identical
points.
so (1λ)2x+5+6λ=0
This has solution λ=4±25
Now we can find the radius
Note R=g2+f2c in x2+y2+2gx+2fy+c=0
R2=(1+3λ)2+(3+λ)21012λ
=10λ2
So R=10|λ|
=10(4+25) or 10(254)
The smaller of these is
10(254)

1087800_1180111_ans_7e05f15ea1b94d9a91ced8c29f15dad0.png

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