The differential equation(1+y2)xdx−(1+x2)ydy=0 Represents a family of:

[d] Given xdx1+x2=ydy1+y2 Integrating we get, 12log(1+x2)=12log(1+y2)+a ⇒1+x2=c(1+y2),c=e2a x2−cy2=c−1⇒x2c−1−y2(c−1c)=1            ? (1) Clearly c>0 as c=e2a Hence, the equation (i) gives a family of hyperbolas with eccentricity =c−1+c−1cc−1−−−−−−−−−−√=c2−1c−1−−−−−−√=c+1−−−−√ if c≠1 Thus eccentricity varies form number to member of the family as it depends on c. If c=1, it is a pair of lines x2−y2=0