If the vertex of the parabola $y = x^2 - 8x + c$ lies on the $x$-axis,then the value of $c$ is:

Let $O$ be the vertex of the parabola $x^{2}=4y$ and $Q$ be any point on it. Let the locus of the point $P$,which divides the line segment $OQ$ internally in the ratio $2:3$,be the conic $C$. Then the equation of the chord of $C$,which is bisected at the point $(1, 2)$,is:

The tangent to the parabola $y^2 = 4ax$ at the point $(a, 2a)$ makes an angle with the $x$-axis equal to

The locus of the poles of focal chords of the parabola $y^2 = 4ax$ is

Let $A$ and $B$ be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $d$ denotes the distance between $A$ and $B$,and $a$ denotes the area of $\triangle SAB$,where $S$ is the focus of the parabola $y^2=4x$,then the value of $(a+d)$ is:

The equation $y^{2}+4x+4y+k=0$ represents a parabola whose latus rectum is