Statement-I The point (5, -3) inside the hyperbola `3x^(2)-5y^(2)+1=0`.
Statement-II The point `(x_1, y_1)` inside the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, then `(x_(1)^(2))/(a^(2))+(y_(1)^(2))/(b^(2))-1lt0`.

Options

A. a) Statement-I is true, Statement-II is also true, Statement-II is the correct explanation of Statement-I.
B. b) Statement-I is true, Statement-II is also true, Statement-II is not the correct explanation of Statement-I.
C. c) Statement-I is true, Statement-II is false.
D. d) Statement-I is false, Statement-II is true

Answer

To solve the problem, we need to analyze both statements given in the question.

### Step 1: Analyze Statement I
We need to check if the point (5, -3) lies inside the hyperbola defined by the equation:

\[ 3x^2 - 5y^2 + 1 = 0 \]

We can rewrite this equation as:

\[ 3x^2 - 5y^2 = -1 \]

### Step 2: Substitute the Point into the Hyperbola Equation
Now, substitute \(x = 5\) and \(y = -3\) into the equation:

\[ 3(5^2) - 5(-3^2) + 1 \]

Calculating each term:

\[ 5^2 = 25 \quad \Rightarrow \quad 3(25) = 75 \]
\[ -3^2 = 9 \quad \Rightarrow \quad -5(9) = -45 \]

Now substituting these values back into the equation:

\[ 75 - 45 + 1 = 75 - 45 + 1 = 31 \]

### Step 3: Determine the Position of the Point
Since \(31 > 0\), we conclude that:

\[ 3(5^2) - 5(-3^2) + 1 > 0 \]

This indicates that the point (5, -3) lies **inside** the hyperbola.

### Step 4: Analyze Statement II
The second statement claims that if a point \((x_1, y_1)\) is inside the hyperbola defined by:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Then:

\[ \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1 < 0 \]

### Step 5: Understanding the Condition
For a point to be inside the hyperbola, the condition is:

\[ \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} < 1 \]

This implies that:

\[ \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1 < 0 \]

This statement is indeed true as it correctly describes the condition for a point being inside the hyperbola.

### Conclusion
- Statement I is **true**: The point (5, -3) is inside the hyperbola.
- Statement II is also **true**: The condition for a point being inside the hyperbola is correctly stated.

### Final Answer
Thus, the correct option is:

**Option B**: Statement 1 is true, statement 2 is also true, but statement 2 is not the correct explanation of statement 1.