Use the Lagrange multiplier method to find the shortest distance from the origin to
the hyperbola .

To find the shortest distance from the origin to the hyperbola defined by , we can apply the Lagrange multiplier method. The distance from the origin can be expressed as . To simplify our calculations, we can minimize instead of .

The constraint we need to satisfy is .

We will set up the system of equations using the Lagrange multipliers :

Now, we can evaluate equations (1) and (2) to express in terms of and and solve the resultant equations. Substitute possible values, starting with and solving case-by-case or substituting to simplify values.

This yields critical points, which we substitute back into equation (3) to find , ensuring to consider all solutions obtained from the multipliers as we could have multiple local minima and maxima.

Following these substitutions and simplifications will eventually lead us to the point(s) where the shortest distance occurs. The process reveals the minimum distance point on the hyperbola accordingly to the distance formula we’ve laid out.

After solving the resulting equations from the above process, one should find the point on the hyperbola nearest to the origin, calculate the corresponding minimum distance . The rigorous checking of all calculations guarantees avoiding common pitfalls, like failing to evaluate all candidates from Lagrange’s setup.

In this case, through adequate calculations, we find that the relevant point on the hyperbola is leading to the shortest distance being .