Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 15] Singular Points of Plane Curves 231
This system has two solutions: 0(0, 0) and N ( --~a t
Oj,'btit the
coordinates of the point N do not satisfy the equation of the given curve.
Hence, there is a unique singular point (0, 0).
Hence,
Fig. 76 Fig. 77 Fig. 78
Let us find the second derivatives and their values at the point 0:
Pig. 79
fa>0
, 4=20,
0=0,
Fig. 80 Fig. 81
if a>0, then A and is an isolated point (Fig. 80);
if a--^0, then The
Aj^O. equation of the curve in this case will be
2 = {/ x 8 or #= Y^\ y exists only when Jc^O; the curve is symmetric
about the x-axis, which is a tangent. Hence, the point M is a cusp of the
first kind (Fig. 81).