1.

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)

Answer»

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,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-2m+n=0`

Solution :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`


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