Parabola

Let$P:y^{2}=4ax,a>0$be a parabola with focus$S$.Let the tangents to the parabola $P$ make an angle of$\frac{\pi\,}{4}$with the line$y=3x+5$touch the parabola$P$at$A$and$B$.Then the value of $a$ for which $A,B$ and$S$are…

Let $P$ be a point on the parabola, $y^{2}=12x$ and $N$ be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point$M$ of$PN$, parallel to its axis which meets…

If the length of thelatus rectum of aparabola, whose focus is $\left(a,a\right)$and the tangent at its vertex is $x+y=a$, is $16$, then$\left|a\right|$is equal to

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin, respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at…

If the tangent to the conic, $y-6=x^{2}$ at $(2,10)$ touches the circle, $x^{2}+y^{2}+8x-2y=k$ (for some fixed$k$) at a point $\left(\alpha\,,\beta\,\right);$ then $(\alpha\,,\beta\,)$ is

Let $A\left(4,-4\right)$ and $B\left(9,6\right)$ be points on the parabola, $y^{2}=4x.$ Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\Delta\,ACB$ is maximum. Then,…

Let $x=2t,y=\frac{t^{2}}{3}$be a conic. Let $S$ be the focus and$B$ be the point on the axis of the conic such that $SA\perp\,BA$, where $A$ is any point on the conic. If $k$is the ordinate of the centroid of the…

A chord is drawn through the focus of the parabola $y^{2}=6x$such that its distance from the vertex of this parabola is $\frac{\sqrt{5}}{2}$, then its slope can be

If $y=m_{1}x+c_{1}$and $y=m_{2}x+c_{2},m_{1}\neq\,m_{2}$are two common tangents of circle $x^{2}+y^{2}=2$and parabola $y^{2}=x$, then the value of $8\left|m_{1}m_{2}\right|$is equal to

Consider the parabola with vertex $\left(\frac{1}{2},\frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}.$Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$…

If the line $ax+y=c$,touches both the curves $x^{2}+y^{2}=1$ and $y^{2}=4\sqrt{2}x$,then $\left|c\right|$ is equal to:

Let the latus rectum of the parabola $y^{2}=4x$ be the common chord to the circles $C_{1}$ and $C_{2}$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles $C_{1}$ and $C_{2}$ is…