Question

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