Question

Give reasons for your answers, either with a short proof or counterexample.? ax^2+by^2+cz^2=d represents a sphere with radius √a^2+b^2+c^2-d, where a,b,c,d are positive real numbers.

Answer 1

See below.

Explanation:

`ax^2+by^2+cz^2=d` represents a sphere only when `a=b=c >0` and `d > 0` or equivalently `a=b=c < 0, d < 0` in this case calling `m = a=b=c` the sphere

`x^2+y^2+z^2=d/m` is origin centered having a radius of `sqrt(d/m)`