Find the point on the hyperbola xy= 8 that is closet to the point (3,0)

The distance between and is:


Since the equation of the hyperbola is , we can get by itself and end up with


which we can plug into our distance formula:


To make calculation easier, we'll square both sides:

and create a new variable :



Differentiate both sides:

Minimum distance is achieved when :





To find a value of , you can use methods like synthetic division and get the answer

Plug into :



So the closest point on the hyperbola to is