Question

Let tangent drawn to the parabola $C_1:y^2=4ax$ at $P$ meets the $y$-axis at $Q$. Another parabola $C_2$ with vertex $Q$ and focus $(0,0)$ is drawn which passes through $(2a,0).$ Identify which of the following statements can be CORRECT?

• A. The equation of the curve $C_2$ is $x^2=4a(y+a)$
• B. The equation of the curve $C_2$ is $x^2=-4a(y+a)$
• C. Point $P$ is the extremity of latus rectum of curve $C_1$
• D. Possible equation of directrix of curve $C_2$ is $y-2a=0$

Answer 1

Answer: (A), (C), (D)

The correct options are
A The equation of the curve $C_2$ is $x^2=4a(y+a)$
C Point $P$ is the extremity of latus rectum of curve $C_1$
D Possible equation of directrix of curve $C_2$ is $y-2a=0$

Tangent at $P(a{t}^2,2at)$ is
$ty=x+a{t}^2$
$Q\equiv (0,at)$

Now, equation of parabola with vertex $(0,at)$ and $(0,0)$ as its focus is $x^2=-4at(y-at)$
The curve passes through $(2a,0)$ $4a^2=-4at(-at)$
$\Rightarrow t^2=1\Rightarrow t=\pm 1$

Possible equation of curve $C_2$ is
$x^2=-4a(y-a)$
or $x^2=4a(y+a)$
Co-ordinates of $P$ are $(a,\pm 2a)$
$[$ Extremities of latus rectum of $C_1]$

Possible equation of directrix of $C_2:$
$y-a=a\Rightarrow y-2a=0$
or $y+a+a=0\Rightarrow y+2a=0$