Equation of a Parabola

Let $PQ$ and $RT$ be two focal chords of the parabola $y^2=16x$. If $P=\left(4,8\right)$ and $R=\left(16,16\right)$ then $QT=$

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

Let $PQ$ be a focal chord of the parabola $y^2=4x$. If the centre of a circle having $PQ$ as its diameter lies on the line $\sqrt{5}y+4=0$, then the length of the chord $PQ$ 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

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

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

The abscissa of a point on the parabola $y^2=20x$ is $7$; find the distance of the point from its focus.

If $t$ is a parameter, then show that $x=2a {\cos }^2 t, y=2a \cos t$ represents the parameteric equation of a parabola.

If the length of a chord intercepted by the parabola $y^2=4ax$ be $4b$, prove that the slope of the chord is $\pm \sqrt{\frac{a}{b-a}}, (b>a)$.

On the parabola $y^2=4ax, P$ is the point with parameter $t, Q$ is the opposite extremity of the focal chord through $P$ and $R$ is the point for which $QR$ is parallel to $PK$ where $K$ is the point $(2a, 0)$. Show that $R$ has parameter $\frac{t^2-1}{t}$.

Two mutually perpendicular chord $OP$ and $OQ$ of the parabola $y^2=4ax$ pass through its vertex $O$. Show that the chord $PQ$ passes through a fixed point of its axis.

Find the equation of the curve represented parametrically as $x=t^2+t+1, y=t^2-t+1$.

The focal distance of a point on the parabola $y^2=8x$ is $4$. Find the co-ordinates of the point.

If $y_1, y_2, y_3$ are the ordinates of three points on the parabola $y^2=4ax$, show that the area of the triangle formed by the points is $\left|\frac{1}{8a}\left(y_2-y_1\right)\left(y_3-y_2\right)\left(y_1-y_3\right)\right|$ sq. unit.