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ILLUSTRATIVE EXAMPLE 2. Show that y = has a salient point at the origin.
Solution. Here 21 =
dx
+
+ e* x(l + e x } 2
If x is positive and approaches zero as a limit, we have ultimately y = Q and ^ = 0. dx t A If x is negative and approaches zero as a limit, we get ultimately y = and = 1. dx Hence at the origin two branches meet, one having OX as its tangent and the other, AB, making an angle of 45 with OX. EXAMPLES . Show that y 2 = 2x 2 + x s has a node at the origin, the slopes of the tangents being V2. . Show that the origin is a node of y 2 (a 2 + x 2 ) = x 2 (a 2 x 2 ), and that the tan- gents bisect the angles between the axes. . Prove that (a, 0) is a node of y 2 = x (x a) 2 , and that the slopes of the tangents are Va. . Prove that a?y 2 2 abx 2 y x 5 = has a point of osculation at the origin. . Show that the curve y 2 = x 5 + x 4 has a point of osculation at the origin. . Show that the cissoid y 2 = has a cusp of the first kind at the origin. a x . Show that y 3 = 2 ox 2 x 3 has a cusp of the first kind at the origin. . In the curve (y x 2 ) 2 = x n show that the origin is a cusp of the first or second kind according as n is < or > 4. . Prove that the curve x 4 2 ox 2 y axy 2 + a 2 y 2 = has a cusp of the second kind at the origin. . Show that the origin is a conjugate point on the curve y 2 (x 2 a 2 ) = x 2 . Show that the curve y 2 = x(a + x) 2 has a conjugate point at ( a, 0). . Show that the origin is a conjugate point on the curve ay 2 x 3 + bx 2 = when a and b have the same sign, and a node when they have opposite signs. . Show that the curve x 4 + 2 ax 2 y ay 3 = has a triple point at the origin, and that the slopes of the tangents are 0, + V2, and V2. . Show that the points of intersection of the curve (-) +(-) =1 with the axes are cusps of the first kind. . Show that no curve of the second or third degree in x and y can have a cusp of the second kind. . Show that y = e * has an end point at the origin. . Show that y = x arc tan - has a salient point at the origin, the slopes of the IT x tangents being
