JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Straight Lines

Consider the set of all lines

p x + q y + r = 0

such that

3 p + 2 q + 4 r = 0

. Which one of the following statements is true?

(A)The lines are concurrent at the point $(\frac{3}{4}, \frac{1}{2})$ .✓ Correct
(B)Each line passes through the origin.
(C)The lines are all parallel
(D)The lines are not concurrent

Answer: (A)

Solution

From

3 p + 2 q + 4 r = 0

solve

r = - \frac{3 p + 2 q}{4}

and substitute into

p x + q y + r = 0

to get

p (x - \frac{3}{4}) + q (y - \frac{1}{2}) = 0

. For this to hold for every allowed

p

and

q

, both brackets must vanish, so every such line passes through the fixed point

(\frac{3}{4}, \frac{1}{2})

and the lines are concurrent there.

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option

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