The length of the latus rectum of the parabola $y^2 - 4y - 2x - 8 = 0$ is:

Suppose a parabola $y=ax^2+bx+c$ has two $x$-intercepts,one positive and one negative,and its vertex is $(2,-2)$. Then,which of the following is true?

The shortest distance between the line $x-y=1$ and the curve $x^{2}=2y$ is .... .

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:

Which of the following lines is a normal to the parabola $y^2 = 16x$?

The points $(at_1^2, 2at_1)$,$(at_2^2, 2at_2)$,and $(a, 0)$ will be collinear,if