Coordinate Geometry

State the coordinates of the centre of the circle in terms of $r$.

Given that the distance between the origin and the centre of the circle is 15 units, find the value of $r$.

The point on the circle furthest from the origin is denoted by $P$.

Find the gradient of the tangent to the circle at $P$.

Find the coordinates of $A$, $B$ and $C$.

Find the angle $ACB$ in radians. Give your answer correct to 3 significant figures.

The chord $AB$ divides the circle into two segments.

Find the area of the larger segment.

Show that one of the possible values of $k$ is 10, and find the other possible value.

It is now given that $k = 10$. A circle passes through the points $P$, $Q$ and $R$.

Find the equation of the tangent to the circle at $R$. Give your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

Find the value of $p$.

Find the coordinates of $C$.

Find the area of the triangle formed by the perpendicular bisector of $AB$ and the $x$- and $y$-axes.

Find an equation of the tangent to the circle at the point $(2, 8)$, giving your answer in the form $ax + by + c = 0$.

Given that the line $x + 3y = k$ does not intersect the circle, show that $k^2 - 20k - 220 > 0$.

Find the coordinates of these two points of intersection.

The circle has centre $C$ and the two points of intersection are denoted by $A$ and $B$.

Find the area of the triangle $ABC$.

Find the equation of the circle.

Find the equation of the tangent to the circle at the point $Q$.

The other point on the circle with $x$-coordinate 7 is $R$.

Find the coordinates of the point of intersection of the tangent at $Q$ with the tangent at $R$.

Find the equation of the circle.

There are two tangents to the circle which have gradient $-\frac{1}{2}$.

Find the exact values of the $x$-coordinates of the points at which these tangents touch the circle.

State the equation of the circle.

The circle passes through the point $(8, k)$.

Find the two possible values of $k$.