The line $y = 6x + 1$ touches the parabola $y^2 = 24x$. The coordinates of a point $P$ on this line,from which the tangent to $y^2 = 24x$ is perpendicular to the line $y = 6x + 1$,is

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15x$ subtends an angle $\theta$ at the vertex of the parabola,then $\sin \frac{\theta}{3}+\cos \frac{2\theta}{3}-\sec \frac{4\theta}{3}=$

If $P_1 P_2$ and $P_3 P_4$ are two focal chords of the parabola $y^2 = 4ax$, then the chords $P_1 P_3$ and $P_2 P_4$ intersect on the

The parabola $y = x^2 + px + q$ cuts the straight line $y = 2x - 3$ at a point with abscissa $1$. If the distance between the vertex of the parabola and the $x$-axis is least,then:

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

If $2x + y + \lambda = 0$ is a focal chord of the parabola $y^2 = -8x$,find the value of $\lambda$.