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Calculus/Min/Max problem

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Expert: - 11/29/2008


Question
Find the point(s) on the hyperbola x²-y²=4 that is closest to the point (1,3).

Answer
Take (x1,y1) to be a point on the curve x²-y²=4.
The distance is √[(x-1)²+(y-3)²].
We can look at the square of the distance since distance is always non-negative and therefore the minimum won't change.  The square of the distance is (x-1)²+(y-3)².

Take the derivative with respect to x of this equation, noting first that you can take (x1,y1) as a point on the curve x²-y²=4.  Using this, y can be solve for and substituted into the equation.

This may make for a little complication in the derivative, but it can be found and set = to 0.

I would get a common denominator then move on part of the derivative to the other side of the equation.  Note that the denominators can then be cancelled and with some squaring, adding, subtracting, dividing, and squarerooting (though not necessarily in that order and possibly many times for each), the equation can be solved.

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