For a parabola with focus $(2, 1)$ and directrix $2x - 3y + 1 = 0$,what is the equation of the latus rectum?

The equation of the diameter of the parabola $y^2 = x$ corresponding to the chord $x - y + 1 = 0$ is

If the line $x + y - 1 = 0$ is tangent to the parabola $y^2 = kx$,find the value of $k$.

$P$ and $Q$ are two distinct points on the parabola,$y^2 = 4x$,with parameters $t$ and $t_1$ respectively. If the normal at $P$ passes through $Q$,then the minimum value of $t_1^2$ is

In the $XY$-plane,three distinct lines $l_1, l_2, l_3$ concur at a point $(\lambda, 0)$. Further,the lines $l_1, l_2, l_3$ are normals to the parabola $y^2=6x$ at the points $A=(x_1, y_1)$,$B=(x_2, y_2)$,and $C=(x_3, y_3)$ respectively. Then,we have:

If $a \ne 0$ and the line $2bx + 3cy + 4d = 0$ passes through the points of intersection of the parabolas $y^2 = 4ax$ and $x^2 = 4ay$,then