STATEMENT-1 : The line `y = (b)/(a)x` will not meet the hyperbola `(x^(2))/(a^(2)) -(y^(2))/(b^(2)) =1, (a gt b gt 0)`.
and
STATEMENT-2 : The line `y = (b)/(a)x` is an asymptote to the hyperbola.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is true, Statement-2 is true , Statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we will analyze the statements regarding the hyperbola and the line.

### Given:
1. Hyperbola equation: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) where \(a > b > 0\)
2. Line equation: \(y = \frac{b}{a}x\)

### Step 1: Identify the asymptotes of the hyperbola
The asymptotes of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) can be found by setting the equation equal to zero:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \]
This simplifies to:
\[ \frac{x^2}{a^2} = \frac{y^2}{b^2} \]
Cross-multiplying gives:
\[ b^2 x^2 = a^2 y^2 \]
Taking the square root of both sides yields:
\[ y = \pm \frac{b}{a} x \]
Thus, the asymptotes are \(y = \frac{b}{a} x\) and \(y = -\frac{b}{a} x\).

### Step 2: Analyze the line \(y = \frac{b}{a} x\)
Since we have found that \(y = \frac{b}{a} x\) is one of the asymptotes of the hyperbola, we can conclude that this line does not intersect the hyperbola at any finite point. Asymptotes are lines that the hyperbola approaches but never intersects.

### Step 3: Conclusion
- **Statement 1**: The line \(y = \frac{b}{a} x\) will not meet the hyperbola. **True**
- **Statement 2**: The line \(y = \frac{b}{a} x\) is an asymptote to the hyperbola. **True**

Since both statements are true and Statement 2 provides a correct explanation for Statement 1, we conclude that both statements are valid.

### Final Answer:
Both statements are true, and Statement 2 is a correct explanation for Statement 1.

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