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Question

If a circle passing through the point (4,3) also touches the lines x+y=2 and xy=2, then which of the following is/are true?

A
The centre of the circle is (10+36,0)
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B
The centre of the circle is (1036,0)
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C
The radius of the circle is r=6233 units
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D
The radius of the circle is r=62+33 units
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Solution

The correct option is D The radius of the circle is r=62+33 units
Given lines are x+y=2 and xy=2 which are perpendicular in nature,


We know that,
r=hk22=h+k22=(h+4)2+(k3)2
Now, using the above conditions
hk22=h+k22
By observation, point (4,3) and centre of the circle lies on the same side of both the lines, so
(hk22)=(h+k22)k=0
Now,
h+k22=(h+4)2+(k3)2h22=(h+4)2+9
Squaring on both sides, we get
(h2)2=2(h2+8h+25)h2+20h+46=0 h=20±4001842h=10±36

So, the centre of the circle are
(10+36,0) and (1036,0)
Therefore,
r=|10+362|2 and r=|10362|2r=|62+33| and r=|6233|r=6233 and r=62+33

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