坐標幾何

If the centre of the circle $x^2+y^2+kx+(k+1)y-3=0$ lies on $x+y+1=0$, find $k$.

If the straight line $y=mx+1$ is tangent to the circle $(x-2)^2+y^2=1$, then $m=$

$A(-1,-4)$ and $B(3,4)$ are two points. The line $x-y=0$ cuts $AB$ at $P$ so that $AP:PB=r:1$. Find $r$.

$A(-4,2)$ and $B(1,-3)$ are two points. $C$ is a point on the $y$-axis such that $AC=CB$. Find the coordinates of $C$.

In the figure, $OABC$ is a parallelogram. If the equation of $OC$ is $2x-y=0$ and the length of $CB$ is $3$, find the equation of $AB$.

In the figure, find the equation of the circle with $AB$ as a diameter.

The figure shows a circle centred at $C$ and passing through $O(0,0)$, $A(6,0)$ and $B(0,8)$. Which of the following must be true? I. $C$ lies on the line $\dfrac{x}{6}+\dfrac{y}{8}=1$. II. The radius of the circle is $10$. III. $OC$ is perpendicular to $AB$.

Two circles with equations $(x+1)^2+(y+1)^2=25$ and $(x-11)^2+(y-8)^2=100$ touch each other externally at a point $P$. Find the coordinates of $P$.

Find the equation of the straight line passing through $(-1,1)$ and parallel to $5x+4y=0$.

In the figure, $PQRS$ is a parallelogram. Find the slope of $PR$.