The length of the chord of the parabola $y^2 = x$ which is bisected at the point $(2, 1)$ is

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$. Find the value of $m$.

$A$ normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$,where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$,then the value of $24a$ is. . . .

Let $A, B$ and $C$ be the vertices of a variable right-angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex containing the right angle be $C = (4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of the latus rectum of $C_o$ is . . . . . .

If two normals are drawn from a point to the parabola $y^{2} = 4ax$ such that they make angles $\alpha$ and $\beta$ with the axis,and $\tan \alpha \cdot \tan \beta = 2$,find the locus of the point.

If the focus and directrix of a parabola are $(3, 5)$ and $x + y = 4$,then the coordinates of its vertex are