a m p ; x 2 + y 2 − 8 x = 0 , x 2 9 − y 2 4 = 1 .... (1) a m p ; 4 x 2 − 9 y 2 = 36 .... (2) a m p ; Solve ( 1 ) & a m p ; ( 2 ) a m p ; 4 x 2 − 9 ( 8 x − x 2 ) = 36 a m p ; 13 x 2 − 72 x − 36 = 0 a m p ; ( 13 x + 6 ) ( x = 6 ) = 0 a m p ; x = − 6 13 , x = 6 a m p ; x = − 6 13 ( rejected ) a m p ; y → Imaginary a m p ; n = 6 , 36 9 − y 2 4 = 1 a m p ; y 2 = 12 , y = I 12 a m p ; A ( 6 , 12 ) , B ( 6 , − 12 ) a m p ; p ( α , 2 α + 4 3 ) P lies on \begin{aligned}& x^2+y^2-8 x=0, \frac{x^2}{9}-\frac{y^2}{4}=1 \quad\text{.... (1)}\\& 4 x^2-9 y^2=36 \quad\text{.... (2)}\\& \text { Solve }(1) \&(2) \\& 4 x^2-9\left(8 x-x^2\right)=36 \\& 13 x^2-72 x-36=0 \\& (13 x+6)(x=6)=0 \\& x=\frac{-6}{13}, x=6 \\& x=\frac{-6}{13}(\text { rejected }) \\& y \rightarrow \text { Imaginary } \\& n=6, \frac{36}{9}-\frac{y^2}{4}=1 \\& y^2=12, y=I \sqrt{12} \\& A(6, \sqrt{12}), B(6,-\sqrt{12}) \\& p\left(\alpha, \frac{2 \alpha+4}{3}\right) P \text { lies on }\end{aligned} am p ; x 2 + y 2 − 8 x = 0 , 9 x 2 − 4 y 2 = 1 .... (1) am p ; 4 x 2 − 9 y 2 = 36 .... (2) am p ; Solve ( 1 ) & am p ; ( 2 ) am p ; 4 x 2 − 9 ( 8 x − x 2 ) = 36 am p ; 13 x 2 − 72 x − 36 = 0 am p ; ( 13 x + 6 ) ( x = 6 ) = 0 am p ; x = 13 − 6 , x = 6 am p ; x = 13 − 6 ( rejected ) am p ; y → Imaginary am p ; n = 6 , 9 36 − 4 y 2 = 1 am p ; y 2 = 12 , y = I 12 am p ; A ( 6 , 12 ) , B ( 6 , − 12 ) am p ; p ( α , 3 2 α + 4 ) P lies on
a m p ; centroid ( h , k ) 2 x − 3 y + y = 0 a m p ; h = 12 + α 3 , α = 3 h − 12 a m p ; k = 2 α − 3 y 3 3 ⇒ 2 α + 4 = 9 y a m p ; α = 9 k − 4 2 a m p ; 6 h − 2 y = 9 k − 4 a m p ; 6 x − 9 y = 20 \begin{aligned}& \text { centroid }(\mathrm{h}, \mathrm{k}) \quad 2x-3y+y=0\\& \mathrm{h}=\frac{12+\alpha}{3}, \alpha=3 \mathrm{~h}-12 \\& \mathrm{k}=\frac{\frac{2 \alpha-3 y}{3}}{3} \Rightarrow 2 \alpha+4=9 \mathrm{y} \\& \alpha=\frac{9 \mathrm{k}-4}{2} \\& 6 \mathrm{~h}-2 \mathrm{y}=9 \mathrm{k}-4 \\& 6 \mathrm{x}-9 \mathrm{y}=20\end{aligned} am p ; centroid ( h , k ) 2 x − 3 y + y = 0 am p ; h = 3 12 + α , α = 3 h − 12 am p ; k = 3 3 2 α − 3 y ⇒ 2 α + 4 = 9 y am p ; α = 2 9 k − 4 am p ; 6 h − 2 y = 9 k − 4 am p ; 6 x − 9 y = 20