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Question

The area of an equilateral triangle inscribed in the circle x2+y2+2gx+2fy+c=0 is

A
332(g2+f2c)
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B
334(g2+f2c)
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C
334(g2+f2+c)
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D
332(g2+f2+c)
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Solution

The correct option is B 334(g2+f2c)
Given circle is x2+y2+2gx+2fy+c=0
Let C be its centre and PQR be an equilateral triangle inscribed in the circle.

C(g,f)
Radius CQ, r=g2+f2c
Since QPR=60QCR=120
( Angle subtended by an arc at the centre of circle is twice the angle subtended at its circumference.)
QCL=12QCR=12×120=60

From QLC,
QL=CQsin60=32g2+f2c
QR=2×QL=3g2+f2c

Now, area of PQR
=34(QR)2
=34×3(g2+f2c)
=334(g2+f2c)

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