The locus of the point of intersection of two prependicular tangents of the ellipse `x^(2)/9+y^(2)/4=1` is

Options

A. `x^(2)+y^(2)=4`
B. `x^(2)+y^(2)=9`
C. `x^(2)+y^(2)=13`
D. `x^(2)+y^(2)=5`

Answer

To find the locus of the point of intersection of two perpendicular tangents of the ellipse given by the equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The given ellipse is in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where:
- \( a^2 = 9 \) (thus \( a = 3 \))
- \( b^2 = 4 \) (thus \( b = 2 \))

### Step 2: Use the formula for the locus of perpendicular tangents
For an ellipse, the locus of the point of intersection of two perpendicular tangents is known as the director circle. The equation of the director circle is given by:
\[ x^2 + y^2 = a^2 + b^2 \]

### Step 3: Calculate \( a^2 + b^2 \)
Substituting the values of \( a^2 \) and \( b^2 \):
\[ a^2 + b^2 = 9 + 4 = 13 \]

### Step 4: Write the equation of the director circle
Thus, the equation of the director circle, which is the locus of the point of intersection of the two perpendicular tangents, is:
\[ x^2 + y^2 = 13 \]

### Conclusion
The locus of the point of intersection of two perpendicular tangents of the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) is given by the equation:
\[ \boxed{x^2 + y^2 = 13} \]

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