Statement I: The conic `sqrtax+sqrtby=1` represents a parabola.
Statement II: Conic `ax^2+2hxy+by^2+2gx+2fy+c=0` represents a parabola if `h^2=ab`.

Options

A. (a) Statement I is true, Statement II is true and Statement II is the correct explanation for Statement I.
B. (b) Statement I is true and Statement II is true but Statement II is not the correct explanation for Statement I.
C. (c) Statement I is false, Statement II is false.
D. (d) Statement I is false, Statement II is true.

Answer

To solve the problem, we need to evaluate the two statements regarding conic sections and determine their validity.

### Step 1: Analyze Statement I
The first statement is:
**"The conic \(\sqrt{ax} + \sqrt{by} = 1\) represents a parabola."**

To analyze this, we can rearrange the equation:
\[ \sqrt{ax} + \sqrt{by} = 1 \]
Squaring both sides gives:
\[ (\sqrt{ax} + \sqrt{by})^2 = 1^2 \]
Expanding the left side:
\[ ax + by + 2\sqrt{abxy} = 1 \]
Rearranging this gives:
\[ 2\sqrt{abxy} = 1 - ax - by \]
This equation does not represent a standard conic section form, and it can be shown that it represents a line rather than a parabola. Therefore, Statement I is **false**.

### Step 2: Analyze Statement II
The second statement is:
**"Conic \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) represents a parabola if \(h^2 = ab\)."**

To determine if this statement is true, we refer to the general condition for a conic section to be a parabola. The condition is that the discriminant \(D\) of the conic must be zero. The discriminant is given by:
\[ D = h^2 - ab \]
For the conic to represent a parabola, we need:
\[ h^2 - ab = 0 \quad \Rightarrow \quad h^2 = ab \]
Since this condition is indeed correct, Statement II is **true**.

### Conclusion
Based on the analysis:
- Statement I is false.
- Statement II is true.

Thus, the correct option is:
**D: Statement I is false, Statement II is true.**