If the normal to the parabola $y^2 = 4ax$ at the point $(a, 2a)$ meets the parabola again at the point $(at^2, 2at)$,then what is the value of $t$?

The locus of the point of intersection of the perpendicular tangents to the parabola $x^2 = 4ay$ is

Let one root of the quadratic equation in $x$: $(k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:

If the chord joining the points $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ on the parabola $y^{2} = 12x$ subtends a right angle at the vertex of the parabola, then $x_{1}x_{2} - y_{1}y_{2}$ is equal to

The distance of the point $(6, 4 \sqrt{3})$ from the focus of the parabola $y^2 = 8x$ is:

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $(2, 3)$ to the parabola $y^2 = 4x$,then what is the value of $\frac{1}{m_1} + \frac{1}{m_2}$?