The lengths of the tangent drawn from any point on the circle 15 x 2 + 15 y 2 − 48 x + 64 y = 0 to the two circles 5 x 2 + 5 y 2 − 24 x + 32 y + 75 = 0 and 5 x 2 + 5 y 2 − 48 x + 64 y + 300 = 0 are in

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The lengths of the tangent drawn from any point on the circle 15 x 2 + 15 y 2 − 48 x + 64 y = 0 to the two circles 5 x 2 + 5 y 2 − 24 x + 32 y + 75 = 0 and 5 x 2 + 5 y 2 − 48 x + 64 y + 300 = 0 are in the ratio of

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Let P ( h , k ) be a point on the circle 15 x 2 + 15 y 2 − 48 x + 64 y = 0 Then the lengths of the tangents from P ( h , k ) to ​ 5 x 2 + 5 y 2 − 24 x + 32 y + 75 = 0 5 x 2 + 5 y 2 − 48 x + 64 y + 300 = 0 are P T 1 ​ = h 2 + k 2 − 5 24 ​ h + 5 32 ​ k + 15 ​ and P T 2 ​ = h 2 + k 2 − 5 48 ​ h + 5 64 ​ k + 60 ​ or P T 1 ​ = 15 48 ​ h − 15 64 ​ k − 5 24 ​ h + 5 32 ​ k + 15 ​ = 15 32 ​ k − 15 24 ​ h + 15 ​ ​ (Since ( h , k ) lies on 15 x 2 − 15 y 2 − 48 x + 64 y = 0 ​ ∴ h 2 + k 2 − 15 48 ​ h + 15 64 ​ k = 0 ) and PT 2 ​ = 15 48 ​ h − 5 64 ​ k − 5 48 ​ h + 5 64 ​ k + 60 ​ = − 15 96 ​ h + 15 128 ​ k + 60 ​ = 2 − 15 24 ​ h + 15 32 ​ k + 15 ​ = 2 PT 1 ​ ⇒ PT 1 ​ : PT 2 ​ = 1 : 2 ​

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