Consider parabola $P_{1} \equiv y = x^{2}$ and $P_{2} \equiv y^{2} = - 8 x$ and the line $L \equiv l x + m y + n = 0$ . Which of the following holds true (a point $(α, β)$ is called rational point if $α$ and $β$ are rational)

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if $l, m, n$ are odd integers then the line L can not intersect parabola $P_{1}$ in a rational point

line L will be tangent to $P_{1}$ if $m, \frac{l}{2}, n$ are in G.P.

if line L is common tangent to $P_{1}$ and $P_{2}$ then $l + m + n = 0$

if line L is common chord of $P_{1}$ and $P_{2}$ then $l - 2 m + n = 0$

answer is A, B, C, D.

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Detailed Solution

Point of intersection of $P_{1}$ and L is given by $m x^{2} + l x + n = 0$

Line is tangent if $l^{2} = 4 m n \Rightarrow m, \frac{l}{2}, n$ are in G.P.
If point of intersection is rational (let $x = \frac{p}{q}$ ) where p and q are co-prime
Then $m p^{2} + l p q + n q^{2} = 0 \dots . . (1)$

Now, if one of p and q is even and other is odd then (1) can not hold as sum of an even and an odd integer can’t be zero
If p, q are odd then (1) can not hold true as sum of three odd numbers can’t be zero
Common tangent to $P_{1}$ and $P_{2}$ is 2x – y – 1 = 0
Common chord of $P_{1}$ and $P_{2}$ is 2x + y = 0