Q1) Prove that the points (a,0),(b,0), and (1,1) are collinear if 1/a + 1/b =1
Q2) Prove that the points (a,b), (c,d), and ( a-c,b-a) are collinear if ad=bc.
Q3) three consecutive vertices of a parallelogram ABCD are A (1,2) B (1,0) and C (4,0).Find the fourth vertex.
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1. If the question is like:
Prove that the points (a, 0), (0, b) and (1, 1) are collinear if , then the solution is:
Let A (a, 0), B (0, b) and C (1, 1) be the given points.
Suppose all given points are collinear.
∴ Area of ∆ABC = 0
⇒ ab – a – b = 0
Dividing both sides by ab, we get
Hence the given points are collinear only if when .
Let the vertex of point D are (x, y).
Since the diagonals of a parallelogram, bisect each other, therefore co-ordinates of mid-point of AC = co-ordinates of mid-point of BD
On comparing equation, we get
⇒ x + 1 = 5 and y = 2
⇒ x = 4 and y = 2
Therefore the vertex of point D are (4, 2).