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
B. 6a
C. 8a
D. 2a

Answer

3

Explanation

Equation of focal chord
`rArr y - y_(1)= m(x-x_(1))`
`rArr y-0 = tan alpha(x-a) rArr y=x tan alpha `
`rArr` Length of focal chord
`=(4)/(m^(2))sqrt(a(I + m^(2))(a-cm))`
`=(4)/(tan^(2) alpha)sqrt(a(I+ tan^(2)alpha)(a+ a tan^(2) alpha))`
`=(4)/(tan^(2) alpha) a(1+ tan^(2) alpha)`
`=4a cosec^(2) alpha" "because alpha in (0, (pi)/(4))`
`therefore ` minimum lenght = 4a(2) = 8a