The focus of the parabola $y = 2x^{2} + x$ is

The area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 20y$ to the ends of its latus rectum is

Study the following statements.
$I$. The vertex of the parabola $x = ly^2 + my + n$ is $\left(n - \frac{m^2}{4l}, -\frac{m}{2l}\right)$.
$II$. The focus of the parabola $y = lx^2 + mx + n$ is $\left(-\frac{m}{2l}, n - \frac{m^2-1}{4l}\right)$.
$III$. The pole of the line $lx + my + n = 0$ with respect to the parabola $x^2 = 4ay$ is $\left(-\frac{2al}{m}, \frac{n}{m}\right)$.
Then,the correct option among the following is:

Let $O$ be the vertex of the parabola $x^{2}=4y$ and $Q$ be any point on it. Let the locus of the point $P$, which divides the line segment $OQ$ internally in the ratio $2:3$, be the conic $C$. Then the equation of the chord of $C$, which is bisected at the point $(1, 2)$, is:

The points $(at_1^2, 2at_1)$,$(at_2^2, 2at_2)$,and $(a, 0)$ will be collinear,if

The equation of the parabola with focus $(a, b)$ and directrix $\frac{x}{a} + \frac{y}{b} = 1$ is given by