Finding the vertex, axis, focus, directrix, and latus rectum of the parabola $\sqrt{x/a}+\sqrt{y/b}=1$ 
Find the vertex, axis, focus, directrix, and length of latus rectum of the parabola 
  $$\displaystyle \sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$$

Try: Curve 
$$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$$ 
represents a parabola.
While drawing the diagram, I am getting that the parabola touches $x$ axis at $A(a,0)$ and touches $y$ axis at $(0,b)$
Also, the tangents at points $A$ and $B$ intersect each other at $(0,0)$ at an angle of $90^\circ$.
I do not know how can I solve this problem. Could someone help me to solve it? Thanks.
 A: A nice property of the parabola states that: the perpendicular from the focus to any tangent intersects it, and the tangent through the vertex, at the same point. Hence, if the tangent at vertex $V$ intersect the axes at $A'=(\alpha,0)$ and $B'=(0,\beta)$, then focus $F$ has coordinates $(\alpha,\beta)$. 
The axis $FV$ is perpendicular to $A'B'$, hence its slope is $\alpha/\beta$. But the tangent at $A$ forms equal angles with the axis and $FA$, hence the slope of line $FA$ is 
$-\alpha/\beta$. The same goes for line $FB$, hence we have the equations:
$$
{\beta-b\over\alpha}={\beta\over\alpha-a}=-{\alpha\over\beta},
$$
which can be solved to find:
$$
\alpha={ab^2\over a^2+b^2},\quad \beta={a^2b\over a^2+b^2}.
$$
The directrix is the parallel to $A'B'$ passing through the origin.

A: The given equation $\displaystyle \sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$
 can be simplified as
$$\displaystyle \sqrt{\frac{x}{a}} = 1 - \sqrt{\frac{y}{b}}$$
Squarring both sides,
$$\displaystyle \frac{x}{a}= 1 - 2\sqrt{\frac{y}{b}} + \frac{y}{b}$$
$$\displaystyle \frac{x}{a}-1+ \frac{y}{b}=- 2\sqrt{\frac{y}{b}}$$
Squarring again,
$$\displaystyle \left(\frac{x}{a}-1+ \frac{y}{b}\right)^2=4\frac{y}{b}$$
$$\displaystyle \left(X\right)^2=4Y$$
Where 
$$X=\frac{x}{a}-1+ \frac{y}{b}$$
and $$Y=\frac{y}{b}$$
Now compare with the standard form, apply the formulae and find the required properties.
Hope this helps you.
