Question

The vertices B and C of  ΔABC lie on the lines 4y 3x = 0 and y = 0, respectively, and the side BC passes through the point P(0, 5). If ABOC is a rhombus where O is the origin and P is an internal point to the rhombus, then the vertex A is

(A)  (3,9)

(B)  (3, -9)

(C)  (3, -9)

(D)  (-3,9)

Answer 1

Correct option  (D)  (-3,9)

Explanation :

See Fig. Let B = (x, 3x/4) and C = (h, 0). Since B, P and C are collinear, we have

Also

When h = 5/3

 Hence, the coordinates of B and C, respectively, are either(4/3,1) and (5/3,0) or (12,9) and (-15,0).

Two cases arise :

Case 1 : B = (4/3,1),P = (0,5) and C = (5/3,0)

 P is an inside point and B, P and C are collinear.

This implies

Therefore,

BP + PC = 9√10 = BC

Hence

A = (12 - 0 - 15,9 - 0 - 0) = (-3,9)