If A and B are the points of intersection of the circle x2 + y2 – 8x = 0 and the hyperbola x29−y24=1 and a point P moves on the line 2x – 3y + 4 = 0, then the centroid of∆PAB lies on the line :

x2+y2−8x=0,x29−y24=1...........(1)4x2−9y2=36............(2)Solve (1) & (2)4x2−98x−x2=3613x2−72x−36=0(13x+6)(x=6)=0x=−613,x=6x=−613 (rejected) y→ Imaginary n=6,369−y24=1y2=12,y=I12A(6,12),B(6,−12)pα,2α+43P lies on centroid (h,k) h=12+α3,α=3h−12k=2α+433⇒2α+4=9kα=9k−426h−24=9y−4=06x−9y=20

If A and B are the points of intersection of the circle x2 + y2 – 8x = 0 and the hyperbola x29−y24=1 and a point P moves on the line 2x – 3y + 4 = 0, then the centroid of∆PAB lies on the line :

x2+y2−8x=0,x29−y24=1...........(1)4x2−9y2=36............(2)Solve (1) & (2)4x2−98x−x2=3613x2−72x−36=0(13x+6)(x=6)=0x=−613,x=6x=−613 (rejected) y→ Imaginary n=6,369−y24=1y2=12,y=I12A(6,12),B(6,−12)pα,2α+43P lies on centroid (h,k) h=12+α3,α=3h−12k=2α+433⇒2α+4=9kα=9k−426h−24=9y−4=06x−9y=20

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If A and B are the points of intersection of the circle $x^{2} + y^{2} - 8 x = 0$ and the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0,$ then the centroid of $∆$ PAB lies on the line :

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4x – 9y = 12

x + 9y = 36

9x – 9y = 32

6x – 9y = 20

answer is D.

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Detailed Solution

$\begin{array}{l} x^{2} + y^{2} - 8 x = 0, \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1 . . . . . . . . . . . (1) \\ 4 x^{2} - 9 y^{2} = 36 . . . . . . . . . . . . (2) \end{array}$

Solve (1) & (2)

$\begin{array}{l} 4 x^{2} - 9 (8 x - x^{2}) = 36 \\ 13 x^{2} - 72 x - 36 = 0 \\ (13 x + 6) (x = 6) = 0 \\ x = \frac{- 6}{13}, x = 6 \\ x = \frac{- 6}{13} (rejected) \\ y \to Imaginary \\ n = 6, \frac{36}{9} - \frac{y^{2}}{4} = 1 \end{array}$

$\begin{array}{l} y^{2} = 12, y = I \sqrt{12} \\ A (6, \sqrt{12}), B (6, - \sqrt{12}) \\ p (α, \frac{2 α + 4}{3}) P lies on \\ centroid (h,k) \\ h = \frac{12 + α}{3}, α = 3 h - 12 \\ k = \frac{\frac{2 α + 4}{3}}{3} \Rightarrow 2 α + 4 = 9 k \\ α = \frac{9 k - 4}{2} \\ 6 h - 24 = 9 y - 4 = 0 \\ 6 x - 9 y = 20 \end{array}$