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  • A line bisecting the ordinate of a point <img alt="\mathrm{P\left(a t^2, 2 a t\right), t>0,}

A line \mathrm{P N} bisecting the ordinate of a point \mathrm{P\left(a t^2, 2 a t\right), t>0,} on the parabola \mathrm{y^2=4 a x} is drawn parallel to the axis to meet the curve at \mathrm{Q}. If \mathrm{N Q} meets the tangents at the vertex at the point \mathrm{T}, then the coordinates of \mathrm{T} are
 

Option: 1

\mathrm{ (0,\frac{4}{3} at )


Option: 2

\mathrm{ (0,2 a t) }


Option: 3

\left(\left (\frac{1}{4} \right )a t^2, a t\right)


Option: 4

\mathrm{(0, a t)}


Answers (1)

best_answer

Equation of the line parallel to the axis and bisecting the ordinate \mathrm{P N} of the point \mathrm{P\left(a t^2, 2 a t\right)} is \mathrm{y=a t} which meet the parabola \mathrm{ y^2=4 a x} at the point \mathrm{ Q\left(\frac{1}{4} a t^2,at \right)}.
Coordinates of N are \mathrm{ \left(a t^2, 0\right).}
Equation of \mathrm{ N Q} is
\mathrm{ y=\frac{0-a t}{a t^2-\frac{1}{4} a t^2} }
which meets the tangent at the vertex, \mathrm{x=0}, at the point \mathrm{y=4 / 3 a t}

.

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vinayak

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