I am confused. How is $y^2 + x^2 =3x$ a circle? Can someone please help me try to understand why the above a circle, or is it just a typo?
Complete the square: $$ x^2 - 3x = \left(x^2 - 3x + \frac 94\right) - \frac 9 4 = \left(x - \frac 3 2\right)^2 - \frac 9 4. $$ Therefore $y^2+x^2-3x=0$ is equivalent to $y^2 + \left(x - \dfrac 3 2 \right)^2 = \dfrac 9 4$.
The purpose of completing the square is always to reduce a problem involving a quadratic polynomial with a first-degree term to a problem involving a quadratic polynomial without a first-degree term.
An equation for a circle (any circle) takes the general form:- $x^2 + y^2 + Dx + Ey + F = 0$.
The characteristics of it is (1) $((x^2)) = ((y^2))$ and (2) ((xy)) = 0; where ((t)) represents the coefficient of the term t.
The equation $y^2 + x^2 = 3x$ meets all of the the above criteria and therefore automatically is an equation of a circle.