The equation of the tangent to the parabola $y^2=8x$ inclined at $30^{\circ}$ to the $X$-axis is

If the equation of a system of parallel chords of the parabola $y^2 = \frac{25x}{7}$ is $4x - y + \lambda = 0$,then the equation of the corresponding diameter is . . . . . .

If a chord,which is not a tangent,of the parabola $y^2=16x$ has the equation $2x+y=p$,and midpoint $(h, k)$,then which of the following is(are) possible value$(s)$ of $p, h$ and $k$?

If the vertex of a parabola is $(0, a)$ and the focus is $(0, 0)$,what is its equation?

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:

$A$ parabola having its axis parallel to the $Y$-axis passes through the points $(0, 2/5)$,$(4, -2)$,and $(1, 8/5)$. Which of the following points lies on this parabola?