If $y = 2x - 3$ is a tangent to the parabola $y^2 = 4a(x - \frac{1}{3})$,then $a$ is equal to:

The vertex and the focus of the parabola $2x^2 + 5y - 6x + 1 = 0$ respectively,are

The length of the normal chord to the parabola $y^2 = 4x$,which subtends a right angle at the vertex,is

If the equation of the directrix of a parabola is $3x + 4y + 15 = 0$ and the equation of the tangent at the vertex is $3x + 4y - 5 = 0$,then what is the length of the latus rectum?

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the endpoints of its latus rectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola,and $\Delta_2$ be the area of the triangle formed by the intersection points of the tangents drawn at $P$ and at the endpoints of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is

The straight line joining any point $P$ on the parabola $y^2 = 4ax$ to the vertex and the perpendicular from the focus to the tangent at $P$ intersect at $R$. Then the equation of the locus of $R$ is: