From an external point $P(h, k)$,a pair of tangent lines are drawn to the parabola $y^2 = 4x$. If $\theta_1$ and $\theta_2$ are the inclinations of these tangents with the $x$-axis such that $\theta_1 + \theta_2 = \frac{\pi}{4}$,then the locus of $P$ is:

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$,be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$,then $12d$ equals ...........

The locus of the vertices of the family of parabolas $6y = 2a^3x^2 + 3a^2x - 12a$ is

The equation of the common tangent to the parabolas $y^2 = 32x$ and $x^2 = 256y$ is

If the parabola $y^2 = 4ax$ passes through the point $(1, -2)$,then the tangent at this point is

If the line $y=x$ is a tangent to the parabola $y=ax^{2}+bx+c$ at the point $(1,1)$ and the curve passes through $(-1,0)$, then