Statement-I The line `4x-5y=0` will not meet the hyperbola `16x^(2)-25y^(2)=400`.
Statement-II The line `4x-5y=0` is an asymptote ot the hyperbola.

Options

A. Statement-I is true, Statement-II is also true, Statement-II is the correct explanation of Statement-I.
B. Statement-I is true, Statement-II is also true, Statement-II is not the correct explanation of Statement-I.
C. Statement-I is true, Statement-II is false.
D. Statement-I is false, Statement-II is true

Answer

To solve the problem, we need to analyze the given statements about the hyperbola and the line. Let's go through the solution step by step.

### Step 1: Write the equation of the hyperbola
The given equation of the hyperbola is:
\[ 16x^2 - 25y^2 = 400 \]

### Step 2: Convert the hyperbola equation to standard form
To convert this equation into standard form, we divide all terms by 400:
\[ \frac{16x^2}{400} - \frac{25y^2}{400} = 1 \]
This simplifies to:
\[ \frac{x^2}{25} - \frac{y^2}{16} = 1 \]
Thus, the standard form of the hyperbola is:
\[ \frac{x^2}{5^2} - \frac{y^2}{4^2} = 1 \]
where \( a = 5 \) and \( b = 4 \).

### Step 3: Find the asymptotes of the hyperbola
The equations of the asymptotes for a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are given by:
\[ \frac{x}{a} - \frac{y}{b} = 0 \quad \text{and} \quad \frac{x}{a} + \frac{y}{b} = 0 \]
Substituting \( a = 5 \) and \( b = 4 \):
1. \( \frac{x}{5} - \frac{y}{4} = 0 \) or \( 4x - 5y = 0 \)
2. \( \frac{x}{5} + \frac{y}{4} = 0 \) or \( 4x + 5y = 0 \)

### Step 4: Analyze the given line
The line given in the problem is:
\[ 4x - 5y = 0 \]
From Step 3, we have established that this line is indeed one of the asymptotes of the hyperbola.

### Step 5: Determine the intersection of the line and the hyperbola
Asymptotes of a hyperbola do not intersect the hyperbola itself. Therefore, the line \( 4x - 5y = 0 \) will not meet the hyperbola.

### Conclusion
- **Statement I**: The line \( 4x - 5y = 0 \) will not meet the hyperbola \( 16x^2 - 25y^2 = 400 \) is **True**.
- **Statement II**: The line \( 4x - 5y = 0 \) is an asymptote of the hyperbola is **True**.

Thus, both statements are true, and Statement II is the correct explanation for Statement I.

### Final Answer
Both statements are true, and Statement II correctly explains Statement I.

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