Coordinate Geometry of the Circle (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

Solutions to this question by accurate drawing will not be accepted. A circle has equation $x^2+y^2–16x–10y+73=0$.

(i) Find the coordinates of the centre of the circle and the length of the radius.

(ii) Hence show that the point (10, 6.5) lies inside the circle.

A different circle has equation ${(x–10)}^2+{(y–6.5)}^2=2.25$.
Show that the two circles touch. You are not required to find the coordinates of the common point.

The circle $C_1$ has a centre at $\left(3,4\right)$ and a radius of $3$.

Write down the equation of $C_1$.

The circle $C_2$ has the equation

$x^2+y^2-12x-16y+100-k^2=0$

where $k>0$.

Given that $C_2$ is tangent to $C_1$, find the value of $k$ in each of the following cases:

(i) The circles touch each other once and neither circle lies inside the other.

(ii) The circles touch each other once and $C_1$ lies inside $C_2$.

A different circle, $C_3$, intersects $C_1$ at two distinct points, one of which has the coordinates $\left(w,7\right)$.

The equation of the common chord to $C_1$ and $C_3$ is $x+y=10$.

Find $w$.

The point $P\left(8,8\right)$ lies on the circle ${\left(x-2\right)}^2+y^2=100$ with centre $C$.

The circle and the tangent to the circle at $P$ are shown below.

Find the equation of the tangent to the circle at $P$, giving your answer in the form $y=mx+c$.

The tangent intersects the $y$-axis at the point $Q$.

Find the area of triangle $PCQ$.

A circle has a centre $(2,–4)$ and radius $3$.
The line$y=2x–3$ intersects the circle at points $A$ and $B$. The perpendicular bisector of line $AB$ intersects the circle at points $X$ and $Y$.

Find the area of kite $AXBY$.