Basics of Parabola

Let $y^2=16x$ be a given parabola and $L$ be an extremity of its latus rectum in the first quadrant. If a chord is drawn through $L$ with slope $-1$, then the length of this chord is

A line bisecting the ordinate $PN$ of a point $P\left(a{t}^2,2at\right), t>0$ on the parabola $y^2=4ax$ is drawn parallel to the axis to meet the curve at $Q$. If $NQ$ meets the tangent at the vertex at the point $T$, then the coordinates of $T$ are

The point on the parabola $y^2=4ax$ nearest to focus has its abscissa

If $P$ be a point on the parabola $y^2=3\left(2x-3\right)$ and $M$ is the foot of perpendicular drawn from $P$ on the directrix of the parabola, then the length of each side of an equilateral triangle $SMP,$ where $S$ is the focus of the parabola is

The length of the latus rectum of the parabola whose focus is $\left(\frac{u^2}{2g}\sin 2\alpha ,-\frac{u^2}{2g}\cos 2\alpha \right)$ and directrix is $y=\frac{u^2}{2g}$ is

Equation of parabola having the extremities of its latus rectum as $\left(3,4\right)$ and $\left(4,3\right)$ is

The point $\left(2a,a\right)$ lies inside the region bounded by the parabola $x^2=4y$ and its latus rectum. Then

Find the length of intercept by the line $4y=3x-48$ on the parabola $y^2=64x$

The equation of the parabola with focus $\left(a,b\right)$ and directrix $\frac{x}{a}+\frac{y}{b}=1$, is given by

The directrix of the parabola $x^2-4x-8y+12=0$ is

The points on the parabola $y^2=12x$ whose focal distance is $4$, are

Through the vertex $O$ (being origin) of the parabola $y^2=4x$, chords $OP$ and $OQ$ are drawn at right angles to one another. The locus of the middle point of $PQ$ is

Let a circle touches the directrix of a parabola $y^2=2ax$ has its centre coinciding with the focus of the parabola. Then the point of intersection of the parabola and circle is

If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then

The equation of parabola whose focus is $\left(5, 3\right)$ and directrix is $3x-4y+1=0$ is

All the three vertices of an equilateral triangle lie on the parabola $y=x^2$ and one of its sides has a slope of $2.$ Then the sum of the $x$-coordinates of the three vertices is

A line from $(-1, 0)$ intersects the parabola $x^2=4y$ at $A$ and $B$. Then the locus of the centroid of $\Delta OAB$ (where $O$ is the origin), is

Let $A \left(0, 2\right), B$ and $C$ be points on parabola $y^2=x+4$ such that $\angle CBA=\frac{\pi }{2}$. Then the range of ordinate of $C$ is

Consider the parabola $x^2+4y=0.$ Let $P(a, b)$ be any fixed point inside the parabola and let $S$ be the focus of the parabola. Then the minimum value of $SQ+PQ$ as point $Q$ moves on the parabola is

Length of the latus rectum of the parabola $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is