An equilateral triangle is inscribed in the parabola $y^2=16ax$ with one of its vertices at the origin. Then,the centroid of that triangle is

The length of the latus rectum of the parabola $4y^{2} + 3x + 3y + 1 = 0$ is

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$. Find the value of $m$.

The parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in $G.P.$,then

Let $Z$ be the point of intersection of the axis and the directrix of the parabola $4x^2 - 12x + 4y + 5 = 0$. If $S$ is its focus,then the point which divides $SZ$ in the ratio $2:1$ 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 ............. $^\circ$.