If a circle with its centre at the focus of the parabola $y^2 = 2px$ is such that it touches the directrix of the parabola,then a point of intersection of the circle and the parabola is

If the normal at point $P(t)$ to the parabola $y^2 = 16x$ meets it again at point $Q(36, -24)$,then the maximum possible focal distance of point $P$ is-

The focal chord to $y^2 = 16x$ is tangent to $(x - 6)^2 + y^2 = 2$. Then,the possible values of the slope of this chord are:

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 ...........

Let $P_1 : y = -x^2 + 4x + 2$ and $P_2 : x^2 + 5x + \frac{17}{8} = y$ be two parabolas. Then,the number of common tangents of $P_1$ and $P_2$ is:

If the focus and directrix of a parabola are $(3, 5)$ and $x + y = 4$,then the coordinates of its vertex are