Question

# Find the ratio of the length of the tangents from any point on the circle 15x2+15y2−48x+64y=0 to the two circles 5x2+5y2−24x+32y+75=0, 5x2+5y2−48x+64y+300=0.1:21:31:42:5

Solution

## The correct option is A 1:2Let  P(h,k) be a point on the circle 15x2+15y2−48x+64y=0. ∴h2+k2−4815h+6415k=0⇒3(h2+k2)=(485h−645k)     ⋯(1) Now let the tangent lengths be PT1 and PT2 from P(h,k) to 5x2+5y2−24x+32y+75=0 and 5x2+5y2−48x+64y+300=0, respectively. Then PT1=√h2+k2−245h+325k+15=√15−12(h2+k2)     [ From (1)] and PT2=√h2+k2−485h+645k+60=√60−2(h2+k2)    [ From (1)]=2√15−12(h2+k2) ∴PT1:PT2=1:2

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