Sketch the function and an example tangent line in the first quadrant.

Label the right-triangle at the corners x0 and y0...

... and at points (x, 0) and (0, 1/x^2) on the axes, that give the x and y values of the point where the tangent touches the curve.

You have 3 similar triangles. So express the equal ratios (that of the big triangle being minus the slope of the tangent) and find x0 and y0 in terms of x.

So get H the hypotenuse in terms of x.

Then see where this function reaches a minimum...