Cartesian equations of circles

problem
Baby circle

Age

16 to 18

Challenge level

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
problem
Ellipses

Age

14 to 18

Challenge level

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.
problem
Square pair circles

Age

16 to 18

Challenge level

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
problem
Curved square

Age

16 to 18

Challenge level

Can you find the area of the central part of this shape? Can you do it in more than one way?
problem
Orthogonal circle

Age

16 to 18

Challenge level

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
problem
Rational round

Age

16 to 18

Challenge level

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.