JEE Main 2021 — Coordinate Geometry

A square $ABCD$ has all its vertices on the curve ${x}^{2}{y}^{2}=1.$ The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is

Choose the correct statement about two circles whose equations are given below: ${x}^{2}+{y}^{2}-10x-10y+41=0$ ${x}^{2}+{y}^{2}-22x-10y+137=0$

Choose the incorrect statement about the two circles whose equations are given below: ${x}^{2}+{y}^{2}-10x-10y+41=0$ and ${x}^{2}+{y}^{2}-16x-10y+80=0$

Consider a circle $C$ which touches the $y-$ axis at $(0,6)$ and cuts off an intercept $6\sqrt{5}$ on the $x-$ axis. Then the radius of the circle $C$ is equal to :

Consider a hyperbola $H:{x}^{2}-2{y}^{2}=4$. Let the tangent at a point $P(4,\sqrt{6})$ meet the $x$-axis at $Q$ and latus rectum at $R({x}_{1},{y}_{1}),{x}_{1}>0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\Delta QFR$ (in sq. units) is equal to

Consider a triangle having vertices $A(-2,3),B(1,9)$ and $C(3,8).$ If a line $L$ passing through the circum-centre of triangle $ABC,$ bisects line $BC,$ and intersects $y$-axis at point $(0,\frac{\alpha }{2}),$ then the value of real number $\alpha$ is _______.

For the four circles $M,N,O$ and $P,$ following four equations are given: Circle $M:{x}^{2}+{y}^{2}=1$ Circle $N:{x}^{2}+{y}^{2}-2x=0$ Circle $O:{x}^{2}+{y}^{2}-2x-2y+1=0$ Circle $P:{x}^{2}+{y}^{2}-2y=0$ If the centre of circle $M$ is joined with centre of the circle $N,$ further centre of circle $N$ is joined with centre of the circle $O,$ centre of circle $O$ is joined with the centre of circle $P$ and lastly, centre of circle $P$ is joined with centre of circle $M,$ then these lines form the sides of a

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $xcosec\alpha -y\mathrm{sec}\alpha =k\mathrm{cot}2\alpha$ and $x\mathrm{sin}\alpha +y\mathrm{cos}\alpha =k\mathrm{sin}2\alpha$ respectively, then ${k}^{2}$ is equal to :

If one of the diameters of the circle ${x}^{2}+{y}^{2}-2x-6y+6=0$ is a chord of another circle $C''$, whose center is at $(2,1)$, then its radius is_____.

If the curve ${x}^{2}+2{y}^{2}=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is

If the curves, $\frac{{x}^{2}}{a}+\frac{{y}^{2}}{b}=1$ and $\frac{{x}^{2}}{c}+\frac{{y}^{2}}{d}=1$ intersect each other at an angle of ${90}^{^{\circ}},$ then which of the following relations is TRUE?

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, ${x}^{2}+{y}^{2}=1$ is a circle of radius $r$, then $r$ is equal to

If the points of intersection of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{{b}^{2}}=1$ and the circle ${x}^{2}+{y}^{2}=4b,b>4$ lie on the curve ${y}^{2}=3{x}^{2}$, then $b$ is equal to :

If the variable line $3x+4y=\alpha$ lies between the two circles $(x-1{)}^{2}+(y-1{)}^{2}=1$ and $(x-9{)}^{2}+(y-1{)}^{2}=4,$ without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is

In a triangle $PQR,$ the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x-y+2=0,$ then the centre of the circumcircle of the $\Delta PQR$ is:

In the circle given below, let $OA=1$ unit, $OB=13$ unit and $PQ\perp OB$. Then, the area of the triangle $PQB$ (in square units) is :

The equation of the circle with centre (1, -2) and radius 3 is:

Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0).$ If the locus of the point $P$ is a circle of radius $r,$ then $4{r}^{2}$ (in the nearest integer) is equal to __________.

Let an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,{a}^{2}>{b}^{2}$, passes through $(\sqrt{\frac{3}{2}},1)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha ,0),\alpha >0$, of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q$, then $P{Q}^{2}$ is equal to :

Let $A(-1,1),B(3,4)$ and $C(2,0)$ be given three points. A line $y=mx,m>0$ , intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let ${A}_{1}$ and ${A}_{2}$ be the areas of $\Delta ABC$ and $\Delta PQC$ respectively, such that ${A}_{1}=3{A}_{2}$, then the value of $m$ is equal to :

Let $A(a,0),B(b,2b+1)$ and $C(0,b),b\neq 0,|b|\neq 1$, be points such that the area of triangle $ABC$ is $1$ sq. unit, then the sum of all possible values of $a$ is:

Let ${r}_{1}$ and ${r}_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle ${x}^{2}+{y}^{2}+2x+4y-4=0$. If $\frac{{r}_{1}}{{r}_{2}}=a+b\sqrt{2},$ then $a+b$ is equal to:

Let $\mathrm{tan}\alpha ,\mathrm{tan}\beta$ and $\mathrm{tan}\gamma ;\alpha ,\beta ,\gamma \neq \frac{(2n-1)\pi }{2},n\in N$ be the slopes of the three line segments $OA,OB$ and $OC,$ respectively, where $O$ is origin. If circumcentre of $\Delta ABC$ coincides with origin and its orthocentre lies on $y$-axis, then the value of ${(\frac{\mathrm{cos}3\alpha +\mathrm{cos}3\beta +\mathrm{cos}3\gamma }{\mathrm{cos}\alpha \cdot \mathrm{cos}\beta \cdot \mathrm{cos}\gamma })}^{2}$ is equal to :

Let $P$ and $Q$ be two distinct points on a circle which has center at $C(2,3)$ and which passes through origin $O.$ If $OC$ is perpendicular to both the line segments $CP$ and $CQ,$ then the set ${P,Q}$ is equal to

Let $A(1,4)$ and $B(1,-5)$ be two points. Let $P$ be a point on the circle ${((x-1))}^{2}+{(y-1)}^{2}=1$, such that ${(PA)}^{2}+{(PB)}^{2}$ have maximum value, then the points, $P,A$ and $B$ lie on

Let ${S}_{1}:{x}^{2}+{y}^{2}=9$ and ${S}_{2}:(x-2{)}^{2}+{y}^{2}=1$. Then the locus of center of a variable circle $S$ which touches ${S}_{1}$ internally and ${S}_{2}$ externally always passes through the points :

Let $A={(x,y)\in R\times R\mid 2{x}^{2}+2{y}^{2}-2x-2y=1}$ $B={(x,y)\in R\times R\mid 4{x}^{2}+4{y}^{2}-16y+7=0}$ and $C={(x,y)\in R\times R\mid {x}^{2}+{y}^{2}-4x-2y+5\leq {r}^{2}}$. Then the minimum value of $|r|$ such that $A\cup B\subseteq C$ is equal to

Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t,0).$ Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y-$axis at $C.$ The locus of the mid-point $P$ of $\mathrm{MC}$ is

Let $P$ be a variable point on the parabola $y=4{x}^{2}+1.$ Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y=x$ is:

Let $B$ be the centre of the circle ${x}^{2}+{y}^{2}-2x+4y+1=0.$ Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1).$ Then $8(\frac{\mathrm{area}\Delta \mathrm{APQ}}{\mathrm{area}\Delta \mathrm{BPQ}})$ is equal to .

Let $A$ be the set of all points $(\alpha ,\beta )$ such that the area of triangle formed by the points $(5,6),$ $(3,2)$ and $(\alpha ,\beta )$ is $12$ square units. Then the least possible length of a line segment joining the origin to a point in $A,$ is :

Let ${E}_{1}:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b.$ Let ${E}_{2}$ be another ellipse such that it touches the end points of major axis of ${E}_{1}$ and the foci of ${E}_{2}$ are the end points of minor axis of ${E}_{1}.$ If ${E}_{1}$ and ${E}_{2}$ have same eccentricities, then its value is:

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\Delta ABC$, then $(R+r)$ is equal to :

Let the circle $S:36{x}^{2}+36{y}^{2}-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle $S,$ then:

Let the equation ${x}^{2}+{y}^{2}+px+(1-p)y+5=0$ represent circles of varying radius $r\in (0,5].$ Then the number of elements in the set $S=${$q:q={p}^{2}$ and $q$ is an integer} is ___________

Let the lengths of intercepts on $x$ -axis and $y$ -axis made by the circle ${x}^{2}+{y}^{2}+ax+2ay+c=0,$ $(a<0)$ be $2\sqrt{2}$ and $2\sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x+2y=0,$ is equal to :

Let the points of intersections of the lines $x-y+1=0,x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle $\mathrm{ABC}.$ Then the area of the triangle $\mathrm{ABC}$ is

The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles ${\mathrm{tan}}^{-1}(\sqrt{2})$ with the straight line, $y+1=3\sqrt{2}x$ is

The intersection of three lines $x-y=0,x+2y=3$ and $2x+y=6$ is a/an

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S(>R)$ respectively from the origin, is :

The line $2x-y+1=0$ is a tangent to the circle at the point $(2,5)$ and the centre of the circle lies on $x-2y=4.$ Then, the radius of the circle is:

The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0),(1,0),(0,1)(1,1)$ is $18$ units, is a circle of diameter $d.$ Then ${d}^{2}$ is equal to

The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$ is :

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16{x}^{2}-9{y}^{2}+32x+36y-164=0$ and its foci is

The locus of the mid-point of the line segment joining the focus of the parabola ${y}^{2}=4ax$ to a moving point of the parabola, is another parabola whose directrix is:

The locus of the point of intersection of the lines $(\sqrt{3})kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4(\sqrt{3})k=0$ is a conic, whose eccentricity is

The minimum distance between any two points ${P}_{1}$ and ${P}_{2}$ while considering point ${P}_{1}$ on one circle and point ${P}_{2}$ on the other circle for the given circles' equations ${x}^{2}+{y}^{2}-10x-10y+41=0$ ${x}^{2}+{y}^{2}-24x-10y+160=0$ is ________

The point $P(a,b)$ undergoes the following three transformations successively: $(a)$ reflection about the line $y=x.$ $(b)$ translation through $2$ units along the positive direction of $x-$ axis. $(c)$ rotation through angle $\frac{\pi }{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$, then the value of $2a+b$ is equal to:

Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0.$ If the equation of one of the diagonals of the parallelogram is $11x+7y=9,$ then other diagonal passes through the point:

Two tangents are drawn from the point $P(-1,1)$ to the circle ${x}^{2}+{y}^{2}-2x-6y+6=0.$ If these tangents touch the circle at points $A$ and $B,$ and if $D$ is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle $ABD$ is equal to: