JEE Main Mathematics — Coordinate Geometry previous year questions with solutions.
Let ${P}_{1}$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and ${P}_{2}$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of ${P}_{2}$ is $x+2y=$ _____.
Let $PQ$ be a focal chord of the parabola ${y}^{2}=4x$ such that it subtends an angle of $\frac{\pi }{2}$ at the point $(3,0)$. Let the line segment $PQ$ be also a focal chord of the ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,{a}^{2}>{b}^{2}$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac{1}{{e}^{2}}$ is equal to
Let the eccentricity of an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$, be $\frac{1}{4}$. If this ellipse passes through the point $(-4\sqrt{\frac{2}{5}},3)$, then ${a}^{2}+{b}^{2}$ is equal to
The equation of a circle with center (2, 3) and radius 5 is:
Let a circle $C:{(x-h)}^{2}+{(y-k)}^{2}={r}^{2},k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$, then the value of $h+k+r$ is equal to _____.
Let the mirror image of a circle ${c}_{1}:{x}^{2}+{y}^{2}-2x-6y+\alpha =0$ in line $y=x+1$ be ${c}_{2}:5{x}^{2}+5{y}^{2}+10gx$ $+10fy+38=0$. If $r$ is the radius of circle ${c}_{2}$, then $\alpha +6{r}^{2}$ is equal to ______
Let the locus of the centre $(\alpha ,\beta ),\beta >0$, of the circle which touches the circle ${x}^{2}+{(y-1)}^{2}=1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y=4$ is
If the circles ${x}^{2}+{y}^{2}+6x+8y+16=0$ and ${x}^{2}+{y}^{2}+2(3-\sqrt{3})x+2(4-\sqrt{6})y=k+6\sqrt{3}+8\sqrt{6}$, $k>0$, touch internally at the point $P(\alpha ,\beta )$, then ${(\alpha +\sqrt{3})}^{2}+{(\beta +\sqrt{6})}^{2}$ is equal to _______.
The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0,x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\frac{1}{2}<a<2$, then $p$ is equal to
If vertex of parabola is $(2,-1)$ and equation of its directrix is $4x-3y=21$, then the length of latus rectum is
Let $H:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1,a>0,b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt{2}+\sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of ${a}^{2}+{b}^{2}$ is equal to ______.
If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is $16$, then $|a|$ is equal to
Let $a>0,b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$. Let ${e}^{'}$ and ${l}^{'}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${e}^{2}=\frac{11}{14}l$ and ${({e}^{'})}^{2}=\frac{11}{8}{l}^{'}$, then the value of $77a+44b$ is equal to
If the line $x-1=0$, is a directrix of the hyperbola $k{x}^{2}-{y}^{2}=6$, then the hyperbola passes through the point
Let ${m}_{1},{m}_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that ${a}^{2}+11a+3({m}_{1}^{2}+{m}_{2}^{2})=220$. If one vertex of the square is $(10(\mathrm{cos}\alpha -\mathrm{sin}\alpha ),10(\mathrm{sin}\alpha +\mathrm{cos}\alpha ))$, where $\alpha \in (0,\frac{\pi }{2})$ and the equation of one diagonal is $(cos\alpha -sin\alpha )x+(\mathrm{sin}\alpha +\mathrm{cos}\alpha )y=10$, then $72({\mathrm{sin}}^{4}\alpha +{\mathrm{cos}}^{4}\alpha )+{a}^{2}-3a+13$ is equal to
Let the hyperbola $H:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ pass through the point $(2\sqrt{2},-2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
The line $y=x+1$ meets the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then ${(3r)}^{2}$ is equal to
Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle ${(x-5)}^{2}+{(y-1)}^{2}=\frac{13}{2}$, then ${r}^{2}$ is equal to
A rectangle $R$ with end points of the one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is _____.
Let the foci of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{7}=1$ and the hyperbola $\frac{{x}^{2}}{144}-\frac{{y}^{2}}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:
A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center
A circle ${C}_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y=2x$ gives a chord $OA$ of a circle ${C}_{1}$. Let ${C}_{2}$ be the circle with $OA$ as a diameter. If the tangent to ${C}_{2}$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA:AP$ is equal to
The set of values of $k$ for which the circle $C:4{x}^{2}+4{y}^{2}-12x+8y+k=0$ lies inside the fourth quadrant and the point $(1,-\frac{1}{3})$ lies on or inside the circle $C$ is
Let the abscissae of the two points $P$ and $Q$ be the roots of $2{x}^{2}-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of ${x}^{2}-sx-q=0$. If the equation of the circle described on $PQ$ as diameter is $2({x}^{2}+{y}^{2})-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.