Statement 1 The equation of the director circle to the ellipse `4x^(2)+9y^(2)=36 is x^(2)+y^(2)=13`
Statement 2 The locus of the point of intersection of perpendicular tangents to an ellipse is called the director circle.

Options

A. Statement I is true, statement II is true: statement II is a correct explanation for statement I
B. Statement I is true, statement II is true, statement II is not a correct explanation for 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 statements regarding the ellipse and the director circle. We will go through each statement step by step.

### Step 1: Identify the given ellipse
The equation of the ellipse is given as:
\[ 4x^2 + 9y^2 = 36 \]

### Step 2: Rewrite the ellipse in standard form
To rewrite the equation in standard form, we divide both sides by 36:
\[ \frac{4x^2}{36} + \frac{9y^2}{36} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

### Step 3: Identify the values of \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 9\) (thus \(a = 3\))
- \(b^2 = 4\) (thus \(b = 2\))

### Step 4: Use the formula for the director circle
The formula for the equation of the director circle for an ellipse is:
\[ x^2 + y^2 = a^2 + b^2 \]
Substituting the values of \(a^2\) and \(b^2\):
\[ x^2 + y^2 = 9 + 4 = 13 \]

### Step 5: Verify Statement 1
Statement 1 claims that the equation of the director circle is:
\[ x^2 + y^2 = 13 \]
From our calculations, this is indeed correct.

### Step 6: Analyze Statement 2
Statement 2 states that the locus of the point of intersection of perpendicular tangents to an ellipse is called the director circle. This is a well-known definition in conic sections.

### Conclusion
Both statements are true:
- Statement 1 is true as we derived the correct equation of the director circle.
- Statement 2 is true as it correctly defines the director circle.

Thus, the answer to the question is that both statements are true, and Statement 2 is a correct explanation of Statement 1.

### Final Answer
Both statements are true.

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