Find the locus of the midpoints of the chords of the parabola $x^2 + 4y = 0$ which pass through its focus.

If the line $y = -x + k$ is normal to the curve $y^2 = 16x$,then $k$ is

If $x = t^2$ and $y = 2t$,then the equation of the normal at $t = 1$ is

$A$ pair of tangents is drawn from an external point $P$ to the parabola $y^2 = 4x$. If $\theta_1$ and $\theta_2$ are the angles made by the tangents with the $x$-axis such that $\theta_1 + \theta_2 = \frac{\pi}{4}$,find the locus of $P$.

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$,such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant,enclosed by the parabola $y^2=x$,the curve $S$,and the lines $x=1$ and $x=4$. Then which of the following statements is (are) True?
$(A) \ (4, \sqrt{3}) \in S$
$(B) \ (5, \sqrt{2}) \in S$
$(C)$ Area of $R$ is $\frac{14}{3}-2 \sqrt{3}$
$(D)$ Area of $R$ is $\frac{14}{3}-\sqrt{3}$

Find the equation of the normal to the parabola $y^2 = 8x$ at the point $(2, 4)$.