If the length of a focal chord of the parabola `y^2=4a x` at a distance `b` from the vertex is `c ,` then prove that `b^2c=4a^3`.

Options

A. `2a^(2)=bc`
B. `a^(3)=b^(2)c`
C. `ac=b^(2)`
D. `b^(2)c=4a^(3)`

Answer

D

Explanation

Let `P(at_(1)^(2), 2at_(1))" and "Q(at_(2)^(2), 2at_(2))` be the end points of a focal chord of the parabola `y^(2)=4ax`. Then,
`PQ=a(t_(2)-t_(1))^(2)`
The equation of PQ is
`(t_(1)+t_(2))y=2x-2a" "[becauset_(1)ty_(2)=-1]`
It is given that PQ=c, and it is at distance b form the vertex.
`:." "a(t_(2)-t_(1))^(2)n =c" and, "|(-2a)/(sqrt((t_(1)-t_(2))^(2)+4))|=b`
`rArr" "(t_(1)-t_(2))^(2)=c/a" and, "(2a)/((t_(1)-t_(2)))=b" "[becauset_(1)t_(2)=-1]`
`rArr" "((2a)/a)^(2)=c/arArr4a^(3)=b^(2)c`