As Husky say's, the Trim tool is the one to use.
kazmer wrote:(BTW, why does Lengthen not work with ellipse segments? Just curious.)
The lengthen tool will work with lines and arc's, for elliptical arc's (segments) you will need to break it down a little, by using the explode tool a couple of times to go from ellipse to Polyline to arc! Ellipses/elliptical arcs have different working parameters involved.
The Trim tool will work on both arcs and elliptical arcs providing they are close enough to intersect with the entity you want to Trim (extend) to.
Of course in the case of your two lines you could just select the white line and drag the Blue anchor point to a new position