<s>[INST] 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.
 [/INST] 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.
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