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Question

Chord of contact (L) from point P on the circle x2+y22x4y20=0 makes an angle of 60 at a point on the circumference of circle. If slope of the chord of contact is 12, then 
  1.  L:x2y=(55+6)2
  2. L:x2y=5562
  3. P(1+25,245)
  4. P(125,2+45)


Solution

The correct options are
A  L:x2y=(55+6)2
B L:x2y=5562
C P(1+25,245)
D P(125,2+45)

Let coordinates of P be (h,k). Then
L:hx+ky(x+h)2(y+k)20=0
L:x(h1)+y(k2)h2k20=0
As h1k2=12,
2h+k=4     (1)

OA=r=5
cos60=rOP
OP=10
100=(h1)2+(k2)2
100=(h1)2+4(h1)2         [Using (1)]
(h1)2=20
h=1±25
(h,k)=(1±25,245)

We have
L:x(h1)+y(k2)h2k20=0
x(h1k2)+y=h+2k+20k2
x2y=2(h+2k+20k2)
Putting the value of (h,k) in equation of L,
x2y=(6±55)2

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