Embibe Experts Solutions for Chapter: Parabola, Exercise 1: JEE Main - 1 February 2023 Shift 2

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

The equations of two sides of a variable triangle are $\mathrm{x}=0$ and $\mathrm{y}=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre 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 $\mathrm{x}+\mathrm{y}+2=0$. If $r$ is the radius of its circumcircle, then $r^2$ is equal to

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 $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16x$. If one of these tangents touches the two curves at $Q$ and $R$, then ${\left(QR\right)}^2$ is equal to

Let $y=f\left(x\right)$ represent a parabola with focus $\left(-\frac{1}{2},0\right)$ and directrix $y=-\frac{1}{2}$. Then $S=\left\{x\in \mathrm{ℝ}:{\tan }^{-1}\left(\sqrt{f\left(x\right)}+{\sin }^{-1}\left(\sqrt{f\left(x\right)+1}\right)\right)=\frac{\pi }{2}\right\}:$

Let $S$ be the set of all $a\in N$ such that the area of the triangle formed by the tangent at the point $P\left(b,c\right),b,c\in N$,  on the parabola $y^2=2ax$ and the lines $x=b,y=0$ is $16{\mathrm{unit}}^2$, then $\sum _{a\in S}a$ is equal to _____ .

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 _____ .