From any point P on the parabola `y^(2)=4ax`, perpebdicular PN is drawn on the meeting it at N. Normal at P meets the axis in G. For what value/values of t, the point N divides SG internally in the ratio 1 : 3, where S is the focus ? a. `sqrt((3)/(5))` b. `-sqrt((5)/(3))` c. `-sqrt((3)/(5))` d. `sqrt((5)/(3))`

Options

A. `sqrt((3)/(5))`
B. `sqrt(-(5)/(3))`
C. `sqrt(-(3)/(5))`
D. `sqrt((5)/(3))`

Answer

B, D

Explanation

2,4

`y^(2)4ax` is the given parabola and N divides SG internally in the ratio 1 : 3.
`S-=(a,0)andN-=(at^(2),0)`
Equation of the normal PG is
`y=-tx+2at+at^(3)`
It meet x-axis at y=0.
`:.tx=2at+at^(3)`
`:.x=2a+at^(2)`
`:.G-=(2a+at^(2),0)`
N divides SG internally in the ratio 1 : 3.
`:.at^(2)=(2a+at^(2)+3a)/(4)`
`rArr3at^(2)=5arArrt=pmsqrt((5)/(3))`