JEE Main Mathematics — Coordinate Geometry previous year questions with solutions.
Let $\mathrm{S}$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x$-axis. Let $\mathrm{C}$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to___________
Suppose $A B$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m} < \sqrt{3}$. If the distance of the chord $\mathrm{AB}$ from the origin is $\mathrm{d}$, then $l \mathrm{~d}^2$ is equal to _______
Let $A(-2,-1),B(1,0),C(\alpha ,\beta )$ and $D(\gamma ,\delta )$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $|\alpha +\beta +\gamma +\delta |$ is equal to ______.
Let $P(\alpha ,\beta )$ be a point on the parabola ${y}^{2}=4x$. If $P$ also lies on the chord of the parabola ${x}^{2}=8y$ whose mid point is $(1,\frac{5}{4})$, then $(\alpha -28)(\beta -8)$ is equal to _______.
Let $A(\alpha ,0)$ and $B(0,\beta )$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x-$axis passes through $P$, then the length of the latus rectum of $E$ is equal to
If one of the diameters of the circle ${x}^{2}+{y}^{2}-10x+4y+13=0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle $C$ is
Let $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ is:
A line passing through the point $A(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15^{\circ}$ in the clockwise direction, then its equation in the new position is
If the line segment joining the points $(5,2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :
If the image of the point $(-4,5)$ in the line $x+2 y=2$ lies on the circle $(x+4)^2+(y-3)^2=r^2$, then $\mathrm{r}$ is equal to:
Let ${e}_{1}$ be the eccentricity of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ and ${e}_{2}$ be the eccentricity of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$, which passes through the foci of the hyperbola. If ${e}_{1}{e}_{2}=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is :
Let a variable line passing through the centre of the circle ${x}^{2}+{y}^{2}-16x-4y=0$, meet the positive co-ordinate axes at the point $A\text{and}B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to
Four distinct points $(2k,3k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :
Let $R$ be a rectangle given by the lines $x=0,x=2$, $y=0$ and $y=5$. Let $A(\alpha ,0)$ and $B(0,\beta ),\alpha \in [0,2]$ and $\beta \in [0,5]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4:1$. Then, the mid-point of $AB$ lies on a
A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects x-axis at Q, then the abscissa of Q is
Consider a circle ${C}_{1}:x{}^{2}+{y}^{2}–4x–2y=\alpha –5.$ Let its mirror image in the line $y=2x+1$ be another circle ${C}_{2}:5{x}^{2}+5{y}^{2}–10fx–10gy+36=0.$ Let $r$ be the radius of ${C}_{2}$ . Then $\alpha +r$ is equal to $________$
The distance between the points (1, 2) and (4, 6) is:
If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse ${x}^{2}+4{y}^{2}=36$ is $r$, then $12{r}^{2}$ is equal to
Consider ellipses ${E}_{k}:k{x}^{2}+{k}^{2}{y}^{2}=1,k=1,2,\ldots ,20$. Let ${C}_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse ${E}_{k}$. If ${r}_{k}$ is the radius of the circle ${C}_{k}$, then the value of $\sum _{k=1}^{20}\frac{1}{{r}_{k}^{2}}$ is
The straight lines ${l}_{1}$ and ${l}_{2}$ pass through the origin and trisect the line segment of the line $L:9x+5y=45$ between the axes. If ${m}_{1}$ and ${m}_{2}$ are the slopes of the lines ${l}_{1}$ and ${l}_{2}$, then the point of intersection of the line $y=({m}_{1}+{m}_{2})x$ with $L$ lies on
Consider the triangles with vertices $A(2,1),B(0,0)$ and $C(t,4),t=[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to ___________.
Let the centre of a circle $C$ be $\alpha ,\beta$ and its radius $r<8$. Let $3x+4y=24$ and $3x–4y=32$ be two tangents and $4x+3y=1$ be a normal to $C$. Then $(\alpha -\beta +r)$ is equal to
If the point $(\alpha ,\frac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the lines $x\mathrm{cos}\theta +y\mathrm{sin}\theta =7,\theta \in (0,\frac{\pi }{2})$ between the co-ordinates axes, then $\alpha$ is equal to
Let $A(0,1),B(1,1)$ and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola ${y}^{2}=4ax$ passing through $D$ is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha }{{\beta }^{2}}$ is equal to