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

A

`x^(2) + y^(2) = 25`

B

`x^(2) + y^(2) = 7`

C

`x^(2) + y^(2) = 49`

D

none of these

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To find the locus of the points of intersection of perpendicular tangents to the hyperbola given by the equation \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \), we can follow these steps: ### Step 1: Identify the values of \(a^2\) and \(b^2\) The given hyperbola can be compared with the standard form of a hyperbola, which is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] From the equation \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \), we can identify: - \( a^2 = 9 \) - \( b^2 = 16 \) ### Step 2: Use the formula for the locus of the intersection of perpendicular tangents The formula for the locus of the points of intersection of perpendicular tangents to the hyperbola is given by: \[ x^2 + y^2 = a^2 - b^2 \] ### Step 3: Substitute the values of \(a^2\) and \(b^2\) Now, substituting the values we found: \[ x^2 + y^2 = 9 - 16 \] This simplifies to: \[ x^2 + y^2 = -7 \] ### Step 4: Rearrange the equation We can rearrange this equation to express it in a standard form: \[ x^2 + y^2 + 7 = 0 \] ### Conclusion Thus, the locus of the points of intersection of the perpendicular tangents to the hyperbola is: \[ x^2 + y^2 + 7 = 0 \]
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