JEE Main 2018 — Coordinate Geometry

The locus of the point of intersection of the lines $\sqrt{2}x-y+4\sqrt{2}k=0$ and $\sqrt{2}kx+ky-4\sqrt{2}=0$ ($k$ is any non-zero real parameter) is

A circle passes through the points $(2,3)$ and $(4$, 5). If its centre lies on the line, $y-4 x+3=0$, then its radius is equal to

Two sets $A$ and $B$ are as under: $A={(a, b)\in R\times R :|a-5|<1 \text{and} |b-5|<1};$ $B={(a, b)\in R\times R:4{(a-6)}^{2}+9{(b-5)}^{2}\leq 36}.$ Then :

In a triangle $A B C$, coordianates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively. Then area of $\triangle A B C$ (in sq. units) is

Let $P$ be a point on the parabola ${x}^{2}=4y$. If the distance of $P$ from the center of the circle ${x}^{2}+{y}^{2}+6x+8=0$ is minimum, then the equation of the tangent to the parabola at $P$ is

If the tangent at $(1,7)$ to the curve ${x}^{2}=y-6$ touch the circle ${x}^{2}+{y}^{2}+16x+12y+c=0$ then the value of $c$ is:

If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is

A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:

The tangent to the circle $C_1: x^2+y^2-2 x-1=0$ at the point $(2,1)$ cuts off a chord of length 4 from a circle $C_2$ whose centre is $(3,-2)$. The radius of $C_2$ is

The foot of the perpendicular drawn from the origin, on the line, $3 x+y=\lambda(\lambda \neq 0)$ is $P$. If the line meets $x$-axis at $A$ and $y$-axis at $B$, then the ratio $B P$ $: P A$ is

Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$, then the equation of the common tangent to the two parabolas is :

If a circle $C$, whose radius is $3$, touches externally the circle ${x}^{2}+{y}^{2}+2x-4y-4=0$ at the point $(2,2)$, then the length of the intercept cut by this circle $C$ on the $x$-axis is equal to

Tangent and normal are drawn at $P(16,16)$ on the parabola ${y}^{2}=16x$, which intersect the axis of the parabola at $A&B$, respectively. If $C$ is the center of the circle through the points $P,A&B$ and $\angle CPB=\theta ,$ then a value of $\mathrm{tan}⁡\theta$ is:

If $\beta$ is one of the angles between the normals to the ellipse ${x}^{2}+3{y}^{2}=9$ at the points $(3\mathrm{cos}\theta ,\sqrt{3}\mathrm{sin}\theta )$ and $(-3\mathrm{sin}\theta ,\sqrt{3}\mathrm{cos}\theta );\theta \in (0,\frac{\pi }{2})$; then $\frac{2\mathrm{cot}\beta }{\mathrm{sin}2\theta }$ is equal to :

The sides of a rhombus $A B C D$ are parallel to the lines, $x-y+2=0$ and $7 x-y+3=0$. If the diagonals of the rhombus intersect at $P(1,2)$ and the vertex $A$ (different from the origin) is on the $y$ axis, then the ordinate of $A$ is

A circle passes through the points $(2,3)$ and $(4,5)$. If its centre lies on the line $y-4x+3=0$, then its radius is equal to :