Coordinate Geometry PYQ — Page 24

Statement 1: The only circle having radius $\sqrt{10}$ and a diameter along line $2 x+y=5$ is $x^2+y^2-6 x$ $+2 y=0$. Statement $2: 2 x+y=5$ is a normal to the circle $x^2+y^2-6 x+2 y=0$.

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

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$ is

The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

The circle $x^2+y^2=4 x+8 y+5$ intersects the line $3 x-4 y=m$ at two distinct points if

Three distinct points A, B and C are given in the 2 - dimensional coordinate plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\frac{1}{3}$. Then the circumcentre of the triangle $A B C$ is at the point

The ellipse $x^2+4 y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is