Parabola

The equations of sides$AB$and$AC$of a triangle$ABC$are$\left(\lambda\,+1\right)x+\lambda\,y=4$and$\lambda\,x+\left(1-\lambda\,\right)y+\lambda\,=0$respectively. Its vertex$A$is on the$y-$axis and its orthocentre…

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are…

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points…

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 $A, B$ and $C$ be three points on the parabola $y^2=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the…

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

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 $P$…

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…

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…

A triangle is formed by the tangents at the point $(2,2)$on the curves $y^{2}=2x$ and $x^{2}+y^{2}=4x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r^{2}$is equal to

Let $r$ be the radius of the circle, which touches x -axis at point $(\mathrm{a}, 0), \mathrm{a} \lt 0$ and the parabola $\mathrm{y}^2=9 \mathrm{x}$ at the point $(4,6)$. Then $r$ is equal to ________

Let a line $y=mx\left(m>0\right)$, intersect the parabola, $y^{2}=x$, at a point $P,$ other than the origin. Let the tangent to it a $P$, meet the $x$-axis at the point $Q.$ If area $\left(\Delta\,OPQ\right)=4$square…