JEE Main Mathematics — Coordinate Geometry previous year questions with solutions.
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:
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 $B$ 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 .
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?
The equation of the circle with centre (1, -2) and radius 3 is:
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
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:
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 _______.
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:
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:
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:
Choose the incorrect statement about the two circles whose equations are given below: ${x}^{2}+{y}^{2}-10x-10y+41=0$ and ${x}^{2}+{y}^{2}-16x-10y+80=0$
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:
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 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_____.
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:
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
A hyperbola passes through the foci of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
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 :
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 ${r}_{1}$ 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 $A(a,0),B(b,2b+1)$ 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:
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 ________