坐標幾何

Find the equation of the straight line which passes through $(3,-1)$ and is perpendicular to $2x-y+1=0$.

$A(-3,2)$ and $B(1,3)$ are two points. $C$ is a point on the $AB$ produced such that $AB:BC=1:2$. Find the coordinates of $C$.

$C_1: x^2+y^2=4$ and $C_2: x^2+y^2=9$ are two circles. A chord $AB$ of $C_2$ touches $C_1$. Find the length of $AB$.

In the figure, the equation of the straight line $L$ is

In the figure, $OA=AB$. If the slope of $AB$ is $m$, find the slope of $OA$.

The table below shows the centres and radii of two circles $C_1$ and $C_2$.

Which of the following may represent the relative positions of $C_1$ and $C_2$ ?

In the figure, the equation of the circle is

The points $A(4,-1)$, $B(-2,3)$ and $C(x,5)$ lie on a straight line. Find $x$.

In the figure, the shaded part is bounded by the axes, the lines $x=3$ and $x+y=5$. Find its area.

$AB$ is a diameter of the circle $x^2+y^2-2x-2y-18=0$. If $A$ is $(3,5)$, then $B$ is