Question

$PN$ is an ordinate of the parabola $y^2=9x$. A straight line is drawn through the mid-point $M$ of $PN$ parallel to the axis of the parabola meeting the parabola at $Q.NQ$ meets the tangent at the vertex $A,$ at a point $T$, then $\frac{AT}{NP}=$

• A. $\frac{3}{2}$
• B. $\frac{4}{3}$
• C. $\frac{2}{3}$
• D. $\frac{3}{4}$

Answer 1

The correct option is C $\frac{2}{3}$

Let Parabola $P(y^2=4ax)$, where $a=\frac{9}{4}$.
$PN$ is an ordinate, Let $P$ be $(a{t}^2,2at)\Rightarrow N(a{t}^2,0)$.
Then Mid Point of $PN$ is $M(a{t}^2,at)$
Line $L$ is drawn from $M(a{t}^2,at)$ parallel to parabola axis is $y=at$, intersects parabola $y^2=4ax$ at $Q(\frac{a{t}^2}{4},at)$.
Line $NQ(y=-\frac{4}{3t}(x-a{t}^2))$ intersects Tangent at Vertex $A(0,0)$[ i.e. tangent $(x=0)$] at $T(0,\frac{4at}{3})$.
Length $AT=\frac{4at}{3}$ and $NP=2at$
$\frac{AT}{NP}=\frac{\frac{4at}{3}}{2at}=\frac{2}{3}$
Hence, Option C.