Basics of Parabola

Find the auxiliary circle for parabola $x^2+4x+4y+16=0$.

If the Cartesian co-ordinates of the point on the parabola ${\mathrm{y}}^2=12\mathrm{x}$ whose parameter is $2$ is $\left(p,q\right) \mathrm{then} p+q=$

Value of $p$ when the parabola $y^2=4px$ passes through the point $\left(3, -2\right)$ is

The equation $y^2-8y-x+19=0$ represents

If the equation of a parabola is given as $5x^2-30x+2y=0$, then find the equation of its directrix. 

Let $A$ and $B$ be two distinct points on the parabola $y^2=4x$. If the circle of radius $2$ having $AB$ as its diameter touches the axis of parabola, then the slope of the line joining $A$ and $B$ can be

If $\left(9a,6a\right)$ is a point in the bounded region formed by parabola $y^2=16x$ and $x=9$, then

The focus of the parabola ${\left(y+1\right)}^2=-8\left(x+2\right)$ is

The focus of the parabola $y^2-4y-x+3=0$ is

The length of the latus rectum of $3x^2-4y+6x-3=0$ is

Let $t_1$ and $t_2$ be the parameters of the end points of a focal chord for the parabola $y^2=4ax$. Then, which of the following is correct?

If $x^2+6x+20y-51=0$, then axis of parabola is

The equation of a parabola which passes through the intersection of a straight line $x+y=0$ and the circle $x^2+y^2+4y=0$ is

The equation of the parabola having vertex and focus are at $\left(0, 0\right)$ and $\left(3, 0\right)$, respectively, is $y^2=kx,$ then the value of $k$ is_____

The length of the latus rectum of $3x^2-4y+6x-3=0$ is

The latus rectum of the parabola $y^2=4ax$, whose focal chord is such that $SP=3$ and is given by

Tangent at the vertex divides the distance between directrix and latusrectum in the ratio

If the two ends of the latus rectum are given. How many parabolas can be drawn?

From any point $P$ on the parabola $x^2=8y$ perpendicular is dropped on directrix at $Q$. Line passing through $Q$ and vertex of parabola again meet the parabola at $R$. If coordinates of $R$ are $\left(-4,2\right)$ then coordinates of $P$ are -

Find the co-ordinates of a point on the parabola $y^2=12x$, whose ordinate is twice of its abscissa.