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JEE Main 2000 Mathematics question on Straight Lines and Pair of Lines

The incentre of the triangle with vertices (1,3)(1, \sqrt{3}), (0,0)(0, 0) and (2,0)(2, 0) is
  1. A.
    (1,32)\left(1, \dfrac{\sqrt{3}}{2}\right)
  2. B.
    (23,13)\left(\dfrac{2}{3}, \dfrac{1}{\sqrt{3}}\right)
  3. C.
    (23,32)\left(\dfrac{2}{3}, \dfrac{\sqrt{3}}{2}\right)
  4. D.
    (1,13)\left(1, \dfrac{1}{\sqrt{3}}\right)
Show answer and solution

Correct option: D

Solution

The triangle has vertices A=(1,√3), B=(0,0), C=(2,0). The side lengths are: a=BC=2, b=CA=√((1-2)²+(√3)²)=√(1+3)=2, c=AB=√(1+3)=2. So it's equilateral with all sides=2.

For an equilateral triangle, the incentre coincides with the centroid: ((1+0+2)/3, (√3+0+0)/3) = (1, √3/3) = (1, 1/√3).

[2000]

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