Mathematics Coordinate Geometry questions from JEE Main 2012.
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 two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and its orthocentre is at $(0,0)$, then the third vertex is
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.
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
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 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
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 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
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
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
If the point $(1, a)$ lies between the straight lines $x+y=1$ and $2(x+y)=3$ then a lies in interval
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
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
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 $m^4+2 m^2=24$.
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 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
If the straight lines $x+3 y=4,3 x+y=4$ and $x+y$ $=0$ form a triangle, then the triangle is