The locus of the points of intersection of perpendicualr tangents to `(x^(2))/(16)-(y^(2))/(9)=1` is

Options

A. `x^(2)-y^(2)=7`
B. `x^(2)-y^(2)=25`
C. `x^(2)+y^(2)=25`
D. `x^(2)+y^(2)=7`

Answer

To find the locus of the points of intersection of perpendicular tangents to the hyperbola given by the equation \(\frac{x^2}{16} - \frac{y^2}{9} = 1\), we can follow these steps:

### Step 1: Identify the values of \(a^2\) and \(b^2\)
From the hyperbola equation \(\frac{x^2}{16} - \frac{y^2}{9} = 1\), we can identify:
- \(a^2 = 16\)
- \(b^2 = 9\)

### Step 2: Use the formula for the director circle
The locus of the points of intersection of perpendicular tangents to a hyperbola 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 \(a^2 - b^2\)
Now, substituting the values of \(a^2\) and \(b^2\) into the equation:
\[ a^2 - b^2 = 16 - 9 = 7 \]

### Step 4: Write the equation of the director circle
Substituting \(a^2 - b^2\) into the director circle equation:
\[ x^2 + y^2 = 7 \]

### Conclusion
Thus, the locus of the points of intersection of perpendicular tangents to the hyperbola is:
\[ \boxed{x^2 + y^2 = 7} \]

---