Find the coordinates of the point of intersection of the tangents at the endpoints of the latus rectum of the parabola $y^2 = 4x$.

If the tangent at a point $P$ on a parabola makes an angle $\alpha$ with its focal distance,then the angle between the tangent and the axis of the parabola is:

What is the area of the triangle formed by the vertex and the endpoints of the latus rectum of the parabola $x^2 = 8y$?

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,passing through $(5, 2)$,and symmetric with respect to the $y$-axis.

Three points $O(0,0)$,$P(a, a^2)$,and $Q(-b, b^2)$ with $a > 0$ and $b > 0$ lie on the parabola $y = x^2$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola,and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $m + n$ is equal to:

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