shortest distance

Monsu

hi, pls anyone, how would i find the shortest distance btw a point (x,y) and a hyperbola , given the equation of the hyperbola??

Tide

Here's one way:

The slope of the line that connects the given point to the one closest on the hyperbola will be the negative reciprocal of the tangent to the point on the hyperbola (i.e. the connecting line and the tangent will be perpendicular to each other).

This should give the same result(s) as explicity writing the distance between the given point and an arbitrary point on the hyperbola them minimizing it.