The equation of the parabola with focus $(1, -1)$ and directrix $x+y+3=0$ 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:

The equation of the parabola with the focus $(3,0)$ and the directrix $x+3=0$ is:

The tangent drawn at any point $P$ to the parabola ${y^2} = 4ax$ meets the directrix at the point $K$. Then the angle which $KP$ subtends at its focus is ............. $^\circ$.

Find the equation of the normal to the curve $x^{2}=4y$ which passes through the point $(1, 2)$.

The parabola with focus at $(4, -3)$ and vertex at $(4, -1)$ is