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Grade: 12
        
Angle between tangentsdrawn from a point p to circle x2+y2-4x-8y+8=0 is 60then length of chord of contact of p is
28 days ago

Answers : (1)

Samyak Jain
325 Points
							
This is a simple question if you know some properties of circle.
I can’t explain it diagrammatically here. Please understand. I’ve explained it fully.
 
Let the centre of circle be O and tangents from P meet the circle at Q and R. Let M be the point of
intersection of QR and OP. Then QR is the chord of contact of P.
 
Radius of circle, r = \sqrt{g^2+f^2-c} = \sqrt{(-2)^2+(-4)^2-8} = \sqrt{12} = 2\sqrt{3}
We know at the point of contact, radius of circle is perpendicular to tangent.
So, \angleOQP = \angleORP = 90\degree.
Also, QR is perpendicular to OP i.e. \angleOMQ = 90\degree.

OP bisects \angleQPR  i.e.  \angleQPO = \angleRPO = 30\degree.
\therefore \angleQOP = 60\degree.
QM = OQsin60\degree = rsin60\degree = 2\sqrt{3}.(\sqrt{3} / 2) = 3.
OP bisects QR i.e. QM = MR  or
QR = 2.QM = 2.3  =  6 unit = length of chord of contact of P.
24 days ago
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