Statement-I The equation of the directrix circle to the hyperbola `5x^(2)-4y^(2)=20` is `x^(2)+y^(2)=1`.
Statement-II Directrix circle is the locus of the point of intersection of perpendicular tangents.

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 two statements regarding the hyperbola given by the equation \(5x^2 - 4y^2 = 20\).

### Step 1: Rewrite the Hyperbola Equation
First, we rewrite the equation of the hyperbola in standard form. The given equation is:
\[ 5x^2 - 4y^2 = 20 \]
Dividing the entire equation by 20, we get:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
From this, we can identify \(a^2 = 4\) and \(b^2 = 5\).

### Step 2: Find the Values of \(a\) and \(b\)
Now, we calculate \(a\) and \(b\):
\[ a = \sqrt{4} = 2, \quad b = \sqrt{5} \]

### Step 3: Use the Formula for the Directrix Circle
The equation of the directrix circle for a hyperbola in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is given by:
\[ x^2 + y^2 = a^2 - b^2 \]
Substituting the values of \(a^2\) and \(b^2\):
\[ x^2 + y^2 = 4 - 5 = -1 \]

### Step 4: Analyze the Result
The equation \(x^2 + y^2 = -1\) does not represent a real circle since the left-hand side (which represents the sum of squares) cannot equal a negative number. Therefore, the equation of the directrix circle does not exist in the real number system.

### Conclusion for Statement I
Since the equation \(x^2 + y^2 = 1\) is not valid, **Statement I is false**.

### Step 5: Evaluate Statement II
Statement II claims that the directrix circle is the locus of the points of intersection of perpendicular tangents to the hyperbola. This is a known property of hyperbolas, and it is indeed true.

### Final Conclusion
- **Statement I** is false.
- **Statement II** is true.

Thus, the correct option is that Statement I is false and Statement II is true.