Consider parabola `P_(1)-=y=x^(2)` and `P_(2)-=y^(2)=-8x` and the line `L-=lx+my+n=0`. Which of the following holds true (a point `(alpha,beta)` is called rational point if `alpha` and `beta` are rational)

Options

A. If `l,m,n` are odd integers then the line `L` can not intersect parabola `P_(1)` in a rational point.
B. Line `L` will be tangent to `P_(1)` if `m,l/2,n` are in G.P.
C. If line `L` is common tangent to `P_(1)` and `P_(2)` then `l+m+n=0`
D. If line `L` is common chord of `P_(1)` and `P_(2)` then `l-2m+n=0`

Answer

A, B, C, D

Explanation

Point of intersection of `P_(1)` and `L` is given by `mx^(2)+lx+n=0`
Line is tangent of `l^(2)=4mnimpliesm,l/2,n` are in G.P.
If point of intersection is rational (let `x=p/q`) where `p` and `q` are co-prime.
Then `mp^(2)+lpq+nq^(2)=0`…….(1)
Now if onne of `p` and `q` is even and other is odd then (1) cannot hold as sum of an even and an odd integer can't be zero.
If `p,q` are odd then (1) cannot hold true as sum of three odd numbers can't be zero.
Common tangent to `P_(1)` and is `2x-y-1=0`
Common chord of `P_(1)` and `P_(2)` is `2x+y=0`