A rod of length eight units moves such that its ends A and B always lie on the lines x − y + 2 = 0 and y + 2 = 0 , respectively. If the locus of the point P , that divides the rod A B internally in th

Question

A rod of length eight units moves such that its ends A and B always lie on the lines x − y + 2 = 0 and y + 2 = 0 , respectively. If the locus of the point P , that divides the rod A B internally in the ratio 2 : 1 is 9 ( x 2 + α y 2 + β x y + γ x + 28 y ) − 76 = 0 , then α − β − γ is equal to :

TestHub · Accepted Answer

​ A B = 8 A B 2 = 64 ​ ⇒ ( a − b ) 2 + ( b + 4 ) 2 = 64 ...(1) Now P divides A B in the ratio 2 : 1 internally ⇒ h = 3 2 a + b ​ and k = 3 − 4 + b + 2 ​ ⇒ 2 a + b = 3 h … (2) k = 3 b − 2 ​ From equation (2) and (3) ⇒ b = 3 k + 2 ​ ⇒ 2 a = 3 h − 3 k − 2 ⇒ a = 2 3 h − 3 k − 2 ​ ​ Now by putting value of a and b in equation ​ ⇒ ( 2 3 h − 3 k − 2 ​ − ( 3 k + 2 ) ) 2 + ( 3 k + 2 + 4 ) 2 = 64 ⇒ ( 2 3 h − 3 k − 2 − 6 k − 4 ​ ) 2 + ( 3 k + 6 ) 2 = 64 ⇒ ( 3 h − 9 k − 6 ) 2 + 4 ( 3 k + 6 ) 2 = 4 × 64 ⇒ 9 ( h − 3 k − 2 ) 2 + 36 ( k + 2 ) 2 = 256 ⇒ 9 ( h 2 + 9 k 2 + 4 − 6 hk − 4 h + 12 k ) + 36 ( k 2 + 4 + 4 k ) = 256 ⇒ 9 ( h 2 + 13 k 2 + 20 − 6 hk − 4 h + 28 k ) = 256 ​ Replacing h by x and k by y ​ ⇒ 9 ( x 2 + 13 y 2 − 6 x y − 4 x + 28 y ) + 180 − 256 = 0 ⇒ 9 ( x 2 + 13 y 2 − 6 x y − 4 x + 28 y ) − 76 = 0 ​ By comparing α = 13 , β = − 6 , γ = − 4 α − β − γ = 13 + 6 + 4 = 23

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