Question

The family of curves represented respectively by the solutions of the differential equations $\frac{dy}{dx}=\frac{x^2+x+1}{y^2+y+1}$ and $\frac{dy}{dx}+\frac{y^2+y+1}{x^2+x+1}=0$

• A. Touch each other
• B. Are orthogonal
• C. Are identical
• D. Neither touch nor intersect

Answer 1

The correct option is B Are orthogonal

For the family of curves represented by the first differential equation the slope of the tangent at any point $C_1(x,y)$ is given by ${\left(\frac{dy}{dx}\right)}_{c_1}=\frac{x^2+x+1}{y^2+y+1}$
For the family of curves represented by the second differential the slope of the tangent at any point is given by
${\left(\frac{dy}{dx}\right)}_{c_1}=-\frac{y^2+y+1}{x^2+x+1}$
Clearly, ${\left(\frac{dy}{dx}\right)}_{c_1}\times {\left(\frac{dy}{dx}\right)}_{c_1}=-1$
Hence, the two curves are orthogonal.