Statement-I Two tangents are drawn from a point on the circle `x^(2)+y^(2)=9` to the hyperbola `(x^(2))/(25)-(y^(2))/(16)=1`, then the angle between tangnets is `(pi)/(2)`.
Statement-II `x^(2)+y^(2)=9` is the directrix circle of `(x^(2))/(25)-(y^(2))/(16)=1`.

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 both statements and verify their correctness step by step.

### Step 1: Identify the hyperbola and its parameters
The given hyperbola is:
\[ \frac{x^2}{25} - \frac{y^2}{16} = 1 \]
From this equation, we can identify:
- \( a^2 = 25 \) (thus \( a = 5 \))
- \( b^2 = 16 \) (thus \( b = 4 \))

### Step 2: Find the equation of the directrix circle
The equation of the directrix circle for a hyperbola is given by:
\[ x^2 + y^2 = a^2 - b^2 \]
Calculating \( a^2 - b^2 \):
\[ a^2 - b^2 = 25 - 16 = 9 \]
Thus, the equation of the directrix circle is:
\[ x^2 + y^2 = 9 \]

### Step 3: Verify Statement II
Statement II claims that \( x^2 + y^2 = 9 \) is the directrix circle of the hyperbola. From our calculation in Step 2, we confirmed that this statement is correct.

### Step 4: Analyze Statement I
Statement I claims that two tangents drawn from a point on the circle \( x^2 + y^2 = 9 \) to the hyperbola will form an angle of \( \frac{\pi}{2} \).

### Step 5: Understand the properties of tangents from the directrix circle
It is a known property that if tangents are drawn from any point on the directrix circle to the hyperbola, the angle between those tangents is \( \frac{\pi}{2} \). Since the circle \( x^2 + y^2 = 9 \) is indeed the directrix circle of the hyperbola, this statement is also correct.

### Conclusion
Both statements are true:
- Statement I is true because the angle between the tangents from the directrix circle to the hyperbola is \( \frac{\pi}{2} \).
- Statement II is true because we have derived the equation of the directrix circle correctly.

Thus, both statements are true, and Statement II correctly explains Statement I.

### Final Answer
Both statements are true, and Statement II is the correct explanation for Statement I.

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