The equation of the tangent to the parabola $y^{2}=4x$ inclined at an angle of $\frac{\pi}{4}$ to the positive direction of $x$-axis is:

The length of the latus-rectum of the parabola whose focus is $\left( \frac{u^2}{2g} \sin 2\alpha, -\frac{u^2}{2g} \cos 2\alpha \right)$ and directrix is $y = \frac{u^2}{2g}$,is

The equation of the parabola whose focus is the point $(0, 0)$ and the tangent at the vertex is $x - y + 1 = 0$ is

Let $\alpha_1$ and $\alpha_2$ be the ordinates of two points $A$ and $B$ on a parabola $y^2=4ax$ and let $\alpha_3$ be the ordinate of the point of intersection of its tangents at $A$ and $B$. Then,$\alpha_3-\alpha_2=$

Find the locus of the midpoints of all focal chords of the parabola $y^2 = 4ax$.

The equation of the line touching both parabolas $y^2=4x$ and $x^2=-32y$ is