JEE Main 2012 — Coordinate Geometry

The point of intersection of the lines $\left(a^3+3\right) x+a y+a-3=0$ and $\left(a^5+2\right) x+(a+2) y+2 a+3=0$ (a real) lies on the $y$-axis for

An ellipse is drawn by taking a diameter of the circle $(x-1)^2+y^2=1$ as its semiminor axis and a diameter of the circle $x^2+(y-2)^2=4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

The line parallel to $x$-axis and passing through the point of intersection of lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$, where $(a, b) \neq(0,0)$ is

If the line $2 x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3: 2$, then $k$ equals

If the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ coincide with the foci of the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$, then $b^2$ is equal to

The number of common tangents of the circles given by $x^2+y^2-8 x-2 y+1=0$ and $x^2+y^2+6 x+8 y=0$ is

The chord $P Q$ of the parabola $y^2=x$, where one end $P$ of the chord is at point $(4,-2)$, is perpendicular to the axis of the parabola. Then the slope of the normal at $Q$ is

If the line $y=m x+1$ meets the circle $x^2+y^2+3 x=0$ in two points equidistant from and on opposite sides of $x$-axis, then

Statement $1$: An equation of a common tangent to the parabola $y^2=16 \sqrt{3} x$ and the ellipse $2 x^2+y^2=4$ is $y=2 x+2 \sqrt{3}$. Statement $2$: If the line $y=m x+\frac{4 \sqrt{3}}{m},(m \neq 0)$ is a common tangent to the parabola $y^2=16 \sqrt{3} x$ and the ellipse $2 x^2+y^2=4$, then $m$ satisfies $m^4+2 m^2=24$.

Let $L$ be the line $y=2 x$, in the two dimensional plane. Statement 1: The image of the point $(0,1)$ in $L$ is the point $\left(\frac{4}{5}, \frac{3}{5}\right)$ Statement 2: The points $(0,1)$ and $\left(\frac{4}{5}, \frac{3}{5}\right)$ lie on opposite sides of the line $\mathrm{L}$ and are at equal distance from it.

If the point $(1, a)$ lies between the straight lines $x+y=1$ and $2(x+y)=3$ then a lies in interval

The area of triangle formed by the lines joining the vertex of the parabola, $x^2=8 y$, to the extremities of its latus rectum is

The length of the diameter of the circle which touches the $x$-axis at the point $(1,0)$ and passes through the point $(2,3)$ is

Statement 1: $y=m x-\frac{1}{m}$ is always a tangent to the parabola, $y^2=-4 x$ for all non-zero values of $m$. Statement 2: Every tangent to the parabola, $y^2=-4 x$ will meet its axis at a point whose abscissa is non-negative.

If the eccentricity of a hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$, which passes through $(K, 2)$, is $\frac{\sqrt{13}}{3}$, then the value of $K^2$ is

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ$, where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and its orthocentre is at $(0,0)$, then the third vertex is

If the straight lines $x+3 y=4,3 x+y=4$ and $x+y$ $=0$ form a triangle, then the triangle is