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Question

Find the ratio of the length of the tangents from any point on the circle 15x2+15y248x+64y=0 to the two circles 5x2+5y224x+32y+75=0, 5x2+5y248x+64y+300=0.
  1. 1:2
  2. 1:3
  3. 1:4
  4. 2:5


Solution

The correct option is A 1:2
Let  P(h,k) be a point on the circle 15x2+15y248x+64y=0.
h2+k24815h+6415k=03(h2+k2)=(485h645k)     (1)
Now let the tangent lengths be PT1 and PT2 from P(h,k) to 5x2+5y224x+32y+75=0 and 5x2+5y248x+64y+300=0, respectively. Then
PT1=h2+k2245h+325k+15=1512(h2+k2)     [ From (1)]
and
PT2=h2+k2485h+645k+60=602(h2+k2)    [ From (1)]=21512(h2+k2)

PT1:PT2=1:2

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