Question

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

• A. touch each other
• B. are orthogonal
• C. are one and the same
• D. none of these

Answer 1

The correct option is B are orthogonal

For the family of curves represented by the first differ-ential equation,
the slope of the tangent at any point $(x,y)$ 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 differ-ential equation,
the slope of the tangent at any point is given by
${\left(\frac{dy}{dx}\right)}_{c_2}=\frac{y^2+y+1}{x^2+x+1}$
Since, ${\left(\frac{dy}{dx}\right)}_{c_1}\times {\left(\frac{dy}{dx}\right)}_{c_2}=-1.$
Hence, the two curves are orthogonal.