Oblique projection for which the projection vector is at an angle of 45 degrees

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dixit:

A special case of oblique projection is called cavalier projection. It is given when the projection vector forms an angle of 45° with the z-axis. This means that: $$(x_p^2+y_p^2)/z_p^2=1$$

My question is: where does this equation comes from??

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The two legs of a $45^\circ$ triangle are equal. The two legs of the triangle in question are $\sqrt{x^2 + y^2}$ and $z$. You can take it from there.