The complex number z is represented by a point M in the argand diagram. Sketch the locus of M if z satisfies the equation "modulus (z-3+4i)=3 times modulus (z-4-3i)".

Can you prove why it is a circle in a geometric way?

I do understand why the resultant form would be a circle in algebraic way.

I can not understand why it is a circle in a geometric way.

Indeed, the reason why I asked this question is that I wonder how to prove that

The loci of point with fixed ratio of distance between two point is a circle.

R1=aR2

The next question I might ask is what would be the loci if the "a" here is a complex number? Why?

If it is a conic section, how what condition or change should be made if other conic section are wanted?