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in Co-ordinate Geometry by (34.1k points)
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If the points P(x, y), A(3, 6) and B (-3, 4) and collinear, then prove that 3x + y – 5 = 0. 

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Let required point P(x, y) which is equidistant from the point A(3, 6) and B(-3, 4)

So, PA = PB

⇒ PA2 = PB2

⇒ (x – 3)2 + (y – 6)2 = [x -(-3)]2 + (y – 4)2

⇒ x2 + 9 – 6x + y2 + 36 – 12y = (x + 3)2 + (y – 4)2

⇒ x2 + y2 – 6x – 12y + 45 = x2 + 6x + 9 + y2 + 16 – 8y2

⇒ x2 + y2 – 6x – 12y + 45 = x2 + y2 + 6x – 8y + 25

⇒ -6x – 12y = 6x – 8y + 25 – 45

⇒ -6x – 12y – 6x + 8y = -20

⇒ -12x – 4y = -20

⇒ 3x + y = 5

⇒ 3x + y – 5 = 0

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