If the line $x + my + am^2 = 0$ is tangent to the parabola $y^2 = 4ax$,find the point of contact.

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $x$-axis is

The smallest value of $x^2 - 3x + 3$ in the interval $(-3, 3/2)$ is

$P$ is a point on the parabola $y^2 = 4ax$ $(a > 0)$ whose vertex is $A$. $PA$ is produced to meet the directrix in $D$ and $M$ is the foot of the perpendicular from $P$ on the directrix. If a circle is described on $MD$ as a diameter,then it intersects the $x$-axis at a point whose coordinates are:

The equation of the normal at the point $\left( \frac{a}{4}, a \right)$ to the parabola $y^2 = 4ax$ is:

If the focal distance of a point on the parabola $y^2 = x$ is $17/4$,and it lies in the first quadrant,find the equation of the normal at this point.