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in Mathematics by (53.3k points)

The vertices B and C of  ΔABC lie on the line 4x − 3y = 0 and x-axis, respectively, BC passes through (2/3, 2/3) and ABOC is a rhombus where O is the origin. Find the equation of the line BC and the coordinates of the vertex A.

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See Fig. OB = OC = CA = AB and the diagonals OA and BC are at right angles. Let OC = a. Suppose B = [x1, 4x1/3]. Equation of the side AB is y = (4x1)/3 and the coordinates of A are [a + x1, (4x1)/3]. Hence

(CA)2 = a2

x12 + 16x12 = a2

a = ± 5x1/3

Case 1: If a  = 5x1/3, we have

and O = (0.0)

By hypothesis, points B, (2/3, 2/3) and C are collinear. This implies that

Now, x1 = 0 = B = (0, 0), which is actually not the origin. Hence, x1 = 3 5/ .Therefore

0 = (0,0), B = (3,5, 4/5), A = 8/5,4/5

 and C = (1,0)

Case 2:  a = -5x1/3

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