Let $\vp_N:S^n\sm\{N\}\rar\R^n$ be the stereographic projection, $y\in S^n$, $y_{n+1}\neq0$ and $H\colon=\{z\in S^{n+1}:\la z,y\ra=0\}$. Verify that $\vp_N(H)$ is the sphere
$$
\{x\in\R^n:\,\sum(x_j+\tfrac{y_j}{y_{n+1}})^2
=\tfrac1{y_{n+1}^2}\}
$$
Let $\psi$ be the inverse of $\vp_N$, then
$$
z=\psi(x)=\left(
\frac{2x_1}{1+\norm x^2},\ldots,\tfrac{2x_n}{1+\norm x^2},
\frac{\norm x^2-1}{1+\norm x^2}\right)~.
$$
As $\la z,y\ra=0$ we infer that:
\begin{eqnarray*}
0&=&\sum_j\frac{2x_jy_j}{1+\norm x^2}
+\frac{(\norm x^2-1)y_{n+1}}{1+\norm x^2}\\
&=&\frac{y_{n+1}}{1+\norm x^2}
\left(\sum(x_j+\frac{y_j}{y_{n+1}})^2
-1-\sum\frac{y_j^2}{y_{n+1}^2}\right)\\
&=&\frac{y_{n+1}}{1+\norm x^2}
\left(\sum(x_j+\frac{y_j}{y_{n+1}})^2
-\frac1{y_{n+1}^2}\right)~.
\end{eqnarray*}