Statement-1: Tangents drawn from any point on the circle `x^(2)+y^(2)=225` to the ellipse `(x^(2))/(144)+(y^(2))/(81)=1` are at a right angle.
Statement -2 : Equation of the auxiliary circle of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 " is " x^(2)+y^(2)=a^(2)`.

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

Explanation

To solve the given statements regarding the tangents drawn from a point on the circle to the ellipse, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
**Statement 1:** Tangents drawn from any point on the circle \(x^2 + y^2 = 225\) to the ellipse \(\frac{x^2}{144} + \frac{y^2}{81} = 1\) are at a right angle.

1. **Identify the Circle and Ellipse:**
- The circle has the equation \(x^2 + y^2 = 225\), which means it has a radius of \(15\) (since \(15^2 = 225\)).
- The ellipse has the equation \(\frac{x^2}{144} + \frac{y^2}{81} = 1\), where \(a^2 = 144\) and \(b^2 = 81\). Thus, \(a = 12\) and \(b = 9\).

2. **Determine the Director Circle of the Ellipse:**
- The equation of the director circle for the ellipse is given by:
\[ x^2 + y^2 = a^2 + b^2 \]
- Substituting the values of \(a^2\) and \(b^2\):
\[ a^2 + b^2 = 144 + 81 = 225 \]
- Therefore, the equation of the director circle is:
\[ x^2 + y^2 = 225 \]

3. **Conclusion for Statement 1:**
- Since the circle \(x^2 + y^2 = 225\) is the same as the director circle of the ellipse, tangents drawn from any point on this circle to the ellipse will indeed be at right angles. Thus, Statement 1 is **true**.

### Step 2: Analyze Statement 2
**Statement 2:** The equation of the auxiliary circle of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is \(x^2 + y^2 = a^2\).

1. **Understanding the Auxiliary Circle:**
- The auxiliary circle of an ellipse is defined as the circle that has the same center as the ellipse and a radius equal to the semi-major axis \(a\).
- The equation of the auxiliary circle is:
\[ x^2 + y^2 = a^2 \]
- This is correct as it represents a circle centered at the origin with radius \(a\).

2. **Conclusion for Statement 2:**
- Since the definition and equation provided in Statement 2 are accurate, Statement 2 is also **true**.

### Final Conclusion
- Both Statement 1 and Statement 2 are true. However, Statement 2 does not provide a correct explanation for Statement 1, as they refer to different concepts (director circle vs. auxiliary circle).

### Answer:
- **Option B:** Statement 1 is true, Statement 2 is true, but Statement 2 is not a correct explanation of Statement 1.