If a focal chord of `y^2=4a x` makes an angle `alpha in [0,pi/4]` with the positive direction of the x-axis, then find the minimum length of this focal chord.

Options

A. `4a sec^(2)alpha`
B. `2a "cosec"^(2)alpha`
C. `4a" cosec"^(2)alpha`
D. `4a cot^(2)alpha`

Answer

C

Explanation

Let `P(at_(1)^(2), 2at_(1))" and Q"(at_(2)^(2), 2at_(2))` be the end points of a focal chord PQ which makes an angle `alpha` with the axis of the parabola. Then,
`PQ=a(t_(2)-t_(1))^(2)`
Also, `tan alpha" = slope of PQ"`
`rArr" "tan alpha=2/(t_(2)+t_(1))rArrt_(1)+t_(2)=2 cot alpha`
`:." "PQ=a(t_(2)-t_(1))^(2)`
`rArr" "PQ=a{(t_(2)+t_(1))^(2)-4t_(1)t_(2)}`
`rArr" "PQ=a {4 cot^(2)alpha+4}" "[becauset_(2)+t_(1)=2 cot alphaandt_(1)t_(2)=-1}`
`rArr" "PQ=4a" cosec"^(2)alpha.`