ISI2015-MMA-47

Question

Consider the family $\mathcal{F}$ of curves in the plane given by $x=cy^2$, where $c$ is a real parameter. Let $\mathcal{G}$ be the family of curves having the following property: every member of $\mathcal{G}$ intersect each member of $\mathcal{F}$ orthogonally. Then $\mathcal{G}$ is given by

• A. $xy=k$
• B. $x^2+y^2=k^2$
• C. $y^2+2x^2=k^2$
• D. $x^2-y^2+2yk=k^2$

Answer 1

lnx = lnc + 2lny

1/x = 2/y * dy/dx

dy/dx = y/2x

dy/dx for orthogonal curve = - 2x/y

This is true for curve (C) as can be seen by differentiation as under

y^2 +2x^2 = k^2

2y dy/dx + 4x = 0

dy/dx = - 2x/y

Answer: (C).