Foot of perpendicular from point P on the parabola `y^(2)=4ax` to the axis is N. A straight line is drawn parallel to the axis which bisects PN and cuts the curve at Q. If NQ meets the tangent at the vertex A at a point T, then `(PN)/(AT)`=__________.

Answer

1.5

Explanation

1.5
Let point P on the parabola be `(at^(2),2at)`.
`:." "PN=2at andN-=(at^(2),0)`
M is midpoint of PN.
Therefore, equation of QM is y=at
Solving this with parabola, we have
`a^(2)t^(2)=4axrArrx=(1)/(4)at^(2)`
`So," "Q-=((1)/(4)at^(2),at)`
Thus, equation of NQ is
`y-0=(at-0)/((1)/(4)at^(2)-at^(2))(x-at^(2))`
`y=-(4)/(3t)(x-at^(2))`
If NQ meets y-axis at `T(0,(4at)/(3))`, then
`AT=(4at)/(3)`
`:." "(PN)/(AT)=(3)/(2)`