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

What is the focus of the parabola $4y^2 + 12x - 20y + 67 = 0$?

The tangents to the parabola $y^2 = 4ax$ from an external point $P$ make angles $\theta_1$ and $\theta_2$ with the axis of the parabola,such that $\tan \theta_1 + \tan \theta_2 = b$,where $b$ is a constant. Then $P$ lies on

The length of the latus rectum of $3x^{2} - 4y + 6x - 3 = 0$ is

The locus of the middle points of all chords of the parabola $y^2 = 4ax$ passing through the vertex is

If a parabolic reflector is $20 \, cm$ in diameter and $5 \, cm$ deep,find the focus.