(120)

To find for the hyperbola , given that the product of the focal distances from a point to the foci is 32, we follow these steps:

Verify the Point on the Hyperbola:

The point satisfies the hyperbola equation:

… (i)

Product of Focal Distances:
The distances from to the foci and are:

Therefore, the product:

Relating , and :
Using the identity for eccentricity:

Substituting in the product equation gives:

…… . (ii)

Solving for and :
From equations (i) and (ii), solve for and :

Calculate :
The conjugate axis length is and the latus rectum is . Thus:

Substituting and gives:

Therefore, .

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