Parabola

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then $\tan\theta=$

A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x$, $0\le y\le6$ at the point $F(x_0,y_0)$. The tangent to the parabola at $F(x_0,y_0)$ intersects the $y$-axis at $G(0,y_1)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum.

Match List I with List II and select the correct answer using the code given below the lists:

Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac12,2\right)$ on the parabola, and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\dfrac{\Delta_1}{\Delta_2}$ is

Let $(x,y)$ be any point on the parabola $y^2=4x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x,y)$ in the ratio $1:3$. Then the locus of $P$ is

Let $L$ be a normal to the parabola $y^2=4x$. If $L$ passes through the point $(9,6)$, then $L$ is given by

Let A and B be two distinct points on the parabola $y^2=4x$. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

The equation of the locus of the point whose distances from the point P and the line AB are equal, is

Match the conics in Column I with the statements/expressions in Column II.

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose

Consider the two curves
$$C_1:y^2=4x,\qquad C_2:x^2+y^2-6x+1=0.$$
Then,

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$, $y_1<0$, $y_2<0$, be the end points of the latus rectum of the ellipse $x^2+4y^2=4$. The equations of parabolas with latus rectum $PQ$ are

The ratio of the areas of the triangles $PQS$ and $PQR$ is

The radius of the circumcircle of the triangle $PRS$ is

The radius of the incircle of the triangle $PQR$ is

STATEMENT-1: The curve $y=\dfrac{-x^2}{2}+x+1$ is symmetric with respect to the line $x=1$.

because

STATEMENT-2: A parabola is symmetric about its axis.