If `A_1B_1` and `A_2B_2` are two focal chords of the parabola `y^2=4a x ,` then the chords `A_1A_2` and `B_1B_2` intersect on

Options

A. directrix
B. axis
C. tangent at vertex
D. none of these

Answer

A

Explanation

(1) `A_(1)B_(1)` is a focal. Then `A_(1)-=(at_(1)^(2),2at_(1))andB_(1)-=(a//t_(1)^(2),-2a//t_(1))`.
`A_(2)B_(2)` is a focal. Then `A_(2)-=(at_(2)^(2),2at_(2))andB_(2)-=(a//t_(2)^(2),-2a//t_(2))`.
Now, the equation of the chord `A_(1)A_(2)` is
`y(t_(1)+t_(2))-2x-2at_(1)t_(2)=0` (1)
Chord `B_(1)B_(2)`
`t(-(1)/(t_(2))-(1)/(t_(2)))-2x-2a(-(1)/(t_(1)))(-(1)/(t_(2)))=0`
`ory(t_(1)+t_(2))+2xt_(1)t_(2)+2a=0` (2)
Subtracting (1) from (2), we get
`2x(t_(1)t_(2)+1)+2a(t_(1)t_(2)+1)+0`
`or(x+a)(1+t_(1)t_(2))=0`
`orx+a=0`
Hence, they interset on the directrix.