Let two parallel lines L₁ and L₂ with positive slope are tangent to the circle C₁: x^2+y^2-2 x-16 y+64=0 . If L₁ is also tangent to the circle C₂: x^2+y^2-2 x+2 y-2=0 and equation of L₂ is a a x-b y+c-a a =0 where a, b, c N , then find the

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Let two parallel lines L₁ and L₂ with positive slope are tangent to the circle C₁: x^2+y^2-2 x-16 y+64=0 . If L₁ is also tangent to the circle C₂: x^2+y^2-2 x+2 y-2=0 and equation of L₂ is a a x-b y+c-a a =0 where a, b, c N , then find the value of (a+b+c) .

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Correct answer: 14

aligned & C₁ centre (1,8), radius 1 & C₂ centre (-1,1), radius 2 aligned Line L₁ is tranverse common tangent of C₁ and C₂ then it cut O ₁, O ₂ internally in the ratio of 1: 2 at point P then aligned & P ( 1+2 1+2 , -1+16 1+2 ) & P (1,5) aligned Equation of L₁ with positive slope is (y-5)=m(x-1) m x-y-m+5=0 From O ₂ , we have | m +1- m +5 ~m ^2+1 |=2 m =2 2 2 2 x-y-2 2 +5=0 Equation of L ₂ 2 2 x - y + k =0 | k +2 2 -5 3 |=2 [distance between parallel lines] k= 6+5-2 2 Equation of L₂ with +ve -intercept aligned & 2 2

Let two parallel lines L₁ and L₂ with positive slope are tangent to the circle C₁: x^2+y^2-2 x-16 y+64=0 . If L₁ is also tangent to the circle C₂: x^2+y^2-2 x+2 y-2=0 and equation of L₂ is a a x-b y+c-a a =0 where a, b, c N , then find the value of (a+b+c) .

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