Exams Joint Entrance Examination Mathematics Straight Lines and Pair of Straight Lines Question 27 Question 27 of 303 Hard MCQ
A rod of length eight units moves such that its ends A A A and B B B always lie on the lines x − y + 2 = 0 x-y+2=0 x − y + 2 = 0 and y + 2 = 0 y+2=0 y + 2 = 0 , respectively. If the locus of the point P P P , that divides the rod A B A B A B internally in the ratio 2 : 1 2: 1 2 : 1
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h = 3 β + α 3 k = − 4 + α + 2 3 α = 3 k + 2 2 β = 3 h − a = 3 h − 3 k − 2 A B = 8 ( α − β ) 2 + ( α + 4 ) 2 = 64 ( 3 k + 2 − ( 3 h − 3 k − 2 2 ) ) 2 + ( 3 k + 2 + 4 ) 2 = 64 ( 9 k − 3 h + 6 ) 2 4 + ( 3 k + 6 ) 2 = 64 9 [ ( 3 k − h + 2 ) 2 + 4 ( k + 2 ) 2 ] = 64 × 4 9 ( x 2 + 13 y 2 − 6 x y − 4 x + 28 y ) = 76 α − β − γ = 13 + 6 + 4 = 23 \begin{gathered} h=\frac{3\beta +\alpha }{3} \\ k=\frac{-4+\alpha +2}{3} \\ \alpha =3k+2 \\ 2\beta =3h-a=3h-3k-2 \\ AB=8 \\ (\alpha -\beta {)}^{2}+(\alpha +4{)}^{2}=64 \\ {(3k+2-(\frac{3h-3k-2}{2}))}^{2}+(3k+2+4{)}^{2}=64 \\ \frac{(9k-3h+6{)}^{2}}{4}+(3k+6{)}^{2}=64 \\ 9[(3k-h+2{)}^{2}+4(k+2{)}^{2}]=64\times 4 \\ 9({x}^{2}+13{y}^{2}-6xy-4x+28y)=76 \\ \alpha -\beta -\gamma =13+6+4=23 \end{gathered} h = 3 3 β + α k = 3 − 4 + α + 2 α = 3 k + 2 2 β = 3 h − a = 3 h − 3 k − 2 A B = 8 ( α − β ) 2 + ( α + 4 ) 2 = 64 ( 3 k + 2 − ( 2 3 h − 3 k − 2 )) 2 + ( 3 k + 2 + 4 ) 2 = 64 4 ( 9 k − 3 h + 6 ) 2 + ( 3 k + 6 ) 2 = 64 9 [( 3 k − h + 2 ) 2 + 4 ( k + 2 ) 2 ] = 64 × 4 9 ( x 2 + 13 y 2 − 6 x y − 4 x + 28 y ) = 76 α − β − γ = 13 + 6 + 4 = 23 Answer the question to unlock