The level curves of z=y/x are hyperbolas? justify
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standard equation of hyperbola
x^2/a^2 -y^2/b^2=1
where a and b are constant
if we find slope of tangents
e.g dy/dx
differentiate
2x/a^2-2ydy/dx/b^2=0
2x/a^2=2y/b^2(dy/dx)
dy/dx=(x/y)b^2/a^2
=k(x/y) {k=b^2/a^2
normal =- dx/dy
=-1/k(y/x)
=p(y/x)
you see normal of hyperbola is directly proportional to z
so,
level of curve z=y/x is hyperbola
x^2/a^2 -y^2/b^2=1
where a and b are constant
if we find slope of tangents
e.g dy/dx
differentiate
2x/a^2-2ydy/dx/b^2=0
2x/a^2=2y/b^2(dy/dx)
dy/dx=(x/y)b^2/a^2
=k(x/y) {k=b^2/a^2
normal =- dx/dy
=-1/k(y/x)
=p(y/x)
you see normal of hyperbola is directly proportional to z
so,
level of curve z=y/x is hyperbola
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