Find equation of the tangent line using implicit differentiation

The curve is $$ \sqrt x+\sqrt y = 1$$ by implicit differentiation, $$ \dfrac{1}{2\sqrt x} + \dfrac{1}{2\sqrt y}\dfrac{dy}{dx}=0$$ meaning that $$ \dfrac{dy}{dx}=-\sqrt{\dfrac{y}{x}}$$
At a point $(x_0,y_0)$ in the curve, the tangent line would be $$ y =-\sqrt{\dfrac{y_0}{x_0}}(x-x_0)+y_0 $$
The $y$-intercept is given when $x=0$, which results in $y=\sqrt{y_0x_0}+y_0$. The $x$-intercept is given when $y=0$, which results in $x=\sqrt{y_0x_0}+x_0$.