Question

A particle is moving in the $x y$-plane along a curve $C$ passing through the point $(3,3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$, then $C$ is a parabola with

• A. length of latus rectum 3
• B. length of latus rectum 6
• C. focus $\left(\frac{4}{3}, 0\right)$
• D. focus $\left(0, \frac{3}{4}\right)$

Answer 1

Answer: (A)

Let Point $P ( x , y )$
$Y-y=y^{\prime}(X-x) $
$Y=0 \Rightarrow X=x-\frac{y}{y^{\prime}}$
$Q\left(x-\frac{y}{y^{\prime}}, 0\right)$
Mid Point of $PQ$ lies on $y$ axis
$ x -\frac{ y }{ y ^{\prime}}+ x =0$
$ y ^{\prime}=\frac{ y }{2 \cdot x } \Rightarrow 2 \frac{ dy }{ y }=\frac{ dx }{ x }$
$2 \ell ny =\ell nx +\ell nk$
$y ^{2}= kx $
It passes through $(3,3) \Rightarrow k =3$
curve $c \Rightarrow y ^{2}=3 x$
Length of L.R.$=3$
Focus $=\left(\frac{3}{4}, 0\right)$