Conic Sections · Question 91

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:

1. x + 9y = 36
2. 9x - 9y = 32
3. 4x - 9y = 12
4. 6x - 9y = 20

Answer

(4) 6x - 9y = 20

Solution

From the hyperbola: y2 = 4x2 - 369. Put in the circle:

x2 + 4x2 - 369 - 8x = 0

13x2 - 72x - 36 = 0

(x - 6)(13x + 6) = 0 x = 6 or x = - 613

x = - 613 is rejected (|x| 3 on the hyperbola).

x = 6 y2 = 144 - 369 = 12 y = 2 3

A(6, 2 3), B(6, -2 3)

Let P(h, k) with 2h - 3k + 4 = 0, and centroid G(X, Y):

X = h + 123, Y = k3 h = 3X - 12, k = 3Y

2(3X - 12) - 3(3Y) + 4 = 0

6X - 9Y = 20

The centroid lies on 6x - 9y = 20.

Answer: (4)