Q.Find the equation of the circle with centre at origin and radius $4$.

A circle is the set of every point in a plane that lies at one fixed distance (the radius $r$) from one fixed point (the centre). When the centre is placed at the origin $O(0,0)$, a general point $P(x,y)$ on the circle satisfies $OP=r$. Squaring this distance-formula statement, $OP^2=x^2+y^2=r^2$, gives the standard equation

This is the simplest possible circle equation and the natural starting point for the other two forms: setting $r=3$ instantly gives $x^2+y^2=9$; conversely, an equation like $x^2+y^2=25$ can be read off immediately as a circle of radius $5$ centred at the origin, with no algebra needed. Every later form of a circle's equation (centre-radius, general, parametric) reduces to this one in the special case where the centre happens to be the origin, which is why it is worth recognising on sight.