How to parametrize the curves

Question

I have to parametrize the curve of intersection of 2 surfaces. The surfaces are:

$$y = x,$$

$$z^2 + y^2 = 25.$$

1. Set the second equation to

$$y^2 = 25 - z^2.$$

Answer 1

According to the second equation,

$$ z^2 + y^2 = 25, $$

one may use a parametrization: for $t \in [0, 2\pi]$,

$$ \begin{aligned} y &= 5\cos{t} \\ z &= 5\sin{t} \\ \end{aligned} $$

Since $y = x$,

$$ \gamma(t) = \left\{ \begin{aligned} x &= 5\cos{t} \\ y &= 5\cos{t} \\ z &= 5\sin{t} \\ \end{aligned} \right.\quad t \in [0, 2\pi]. $$