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JEE Main 2019 · 9 Jan · Shift I

JEE Main 2019 Mathematics question on Straight Lines and Pair of Lines

Consider the set of all lines px+qy+r=0px + qy + r = 0 such that 3p+2q+4r=03p + 2q + 4r = 0. Which one of the following statements is true?
  1. A.
    Each line passes through the origin.
  2. B.
    The lines are concurrent at the point (34,12)\left(\dfrac{3}{4}, \dfrac{1}{2}\right)
  3. C.
    The lines are all parallel
  4. D.
    The lines are not concurrent
Show answer and solution

Correct option: B

Solution

From 3p + 2q + 4r = 0, divide by 4: (3/4)p + (1/2)q + r = 0, which can be written as p(3/4) + q(1/2) + r = 0. This means the point (3/4, 1/2) satisfies px + qy + r = 0 for every choice of p, q, r satisfying the constraint. Hence all such lines pass through the fixed point (3/4, 1/2), i.e., they are concurrent at that point.

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