Can you find the equation of a circle which touches the parabola maths equation, but does not cross the parabola?
From symmetry, the centre of the circle must lie on the y axis.
Let the centre of the circle be maths
Equation of circle is maths equation
Equation of parabola is maths equation
At the point of intersection, substituting for y: maths equation
Collecting like terms and simplifying: maths equation
Since the circle touches but does not cross the parabola, this has only one root, so maths equation
The equation of the circle is: maths equation