Basics of Parabola

The extreme points of the laus rectum of the parabola $\left(7,5\right)$ and $\left(7,3\right)$. Find the equation of the parabola.

Let $ABCD$ be a square of side length $2$ units. A line $M$ through $A$ is drawn parallel to $BD$. Point $S$ moves such that its distances from the line $BD$ and the vertex $A$ are equal. If locus of $S$ cuts $M$ at $T_2$ and $T_3$ and $AC$ at $T_1$, then area of $\Delta T_1{T}_2{T}_3$ is

Locus of point of intersection of the straight lines (where $t$ is a real variable parameter) $x-\left(t+1\right)y-\left(t-1\right)=0 \& x-ty-2\left(t-1\right)=0$is

A variable parabola is drawn to pass through $A$ and $B$ ends of diameters of a given circle centred at origin and radius $c$ and to have as directrix a tangent to concentric circle of a radius $a$ $(a>c)$. The coordinate axis being $AB$ and perpendicular diameter. Find the locus of focus of the parabola

From the vertex $O$ of the parabola $y^2=4x$, two mutually perpendicular chords $OP$ and $OQ$ are drawn. Rectangle $OPRQ$ is completed, then locus of point $R$ will be a conic $S=0$ such that

The slope of the line which belongs to family of lines $\left(1+\lambda \right)x+\left(\lambda -1\right)y+2\left(1-\lambda \right)=0$ and makes shortest intercept on $x^2=4y-4$, is

Let the parabolas $y=x^2+ax+b$ and $y=x\left(c-x\right)$ touch each other at $\left(1,0\right)$. Then

If point $(4,4)$ lies on parabola whose focus lies on $x$-axis and directrix is $lx+my=1$ such that $(l,\mathrm{m})$ lies on curve $x^2+y^2-32xy+8x+8y-1=0$ then:

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 $\left(p,q\right) \mathrm{then} p+q=$

If from the vertex of a parabola, a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be made, prove that the locus of the further angle of the rectangle is the parabola $y^2=4a\left(x-8a\right).$

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

Prove that the curve whose polar equation is $r {\cos }^2\frac{\theta }{2}=1$ is a parabola.

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

The parabola $y^2=px$ passes through $\left(4, -2\right)$ Find the length of latus rectum.

Examine whether the point $\left(3, 2\right)$ lies inside, outside or upon the parabola $y^2=6x$.

$\mathrm{P}\left(2t, t^2\right)$ is a moving point. Find the locus of $P$. 

Find the co-ordinates of vertex and the length of latus rectum of the parabola $4y^2+4y+3x-2=0$

Find the locus of the point $(x, y)$ where $x=2t^2+3y=4t+5, t$ being a parameter.

Prove that the locus of the centre of a circle which has an intercept of a given length $2a$ on the axis of $x$ and passes through a given point on the $y$-axis at distance $b$ from the origin is the parabola $x^2-2by+b^2=a^2$.