Parabola

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,…

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

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 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 $x$-intercept of a focal chord of the parabola $y^{2}=8x+4y+4$is $3$, then the length of this chord is equal to _____ .

Let a line perpendicular to the line $2 x-y=10$ touch the parabola $y^2=4(x-9)$ at the point $P$. The distance of the point $P$ from the centre of the circle $x^2+y^2-14 x-8 y+56=0$ is __________

The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are…

The parabolas : $ax^{2}+2bx+cy=0$and $d^{2}+2ex+fy=0$intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in $G.P.$, then

If the shortest distance of the parabola $y^{2}=4x$from the centre of the circle $x^{2}+y^{2}-4x-16y+64=0$is $d$, then $d^{2}$is equal to :

The distance of the point $\left(6,-2\sqrt{2}\right)$from the common tangent $y=mx+c,m>0$, of the curves $x=2y^{2}$and $x=1+y^{2}$is

If $P(h,k)$ be point on the parabola $x=4y^{2}$, which is nearest to the point $Q(0,33)$, then the distance of $P$ from the directrix of the parabola $y^{2}=4(x+y)$ is equal to:

Let the common tangents to the curves $4\left(x^{2}+y^{2}\right)=9$and $y^{2}=4x$intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$and…