If $lx + my + n = 0$ is tangent to the parabola $x^2 = y$,then the condition of tangency is

If the vertex of the conic $y^{2}-4y=4x-4a$ always lies between the straight lines $x+y=3$ and $2x+2y-1=0$, then:

Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(17, 289), (-2, 4), (13, 169)$,then the sum of the perpendicular distances from the directrix of $P$ to all four of the intersection points is:

The point on the parabola $y^2 = 8x$ at which the normal is inclined at $60^\circ$ to the $x$-axis has the coordinates:

The tangents to the parabola $y^2 = 4ax$ from an external point $P$ make angles $\theta_1$ and $\theta_2$ with the axis of the parabola,such that $\tan \theta_1 + \tan \theta_2 = b$,where $b$ is a constant. Then $P$ lies on

Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8x$ to another point on it. Then, the locus of $P$ is