Question

The differential equation of all conics whose axes coincide with the co-ordinate axes, is

• A. $xy{y}_2+x{y}_1^2-y{y}_1=0$
• B. $y{y}_2+y_1^2-y{y}_1=0$
• C. $xy{y}_2+\left(x-y\right)y_1=0$
• D. ${\left(y{y}_1\right)}^2-x{y}_1-yx=0$

Answer 1

The correct option is B $y{y}_2+y_1^2-y{y}_1=0$

Let the equation of the conic be
$\begin{array}{c}a{x}^2+b{y}^2+c=0\\ x^2+\frac{b}{a}{y}^2+\frac{c}{a}=0\end{array}$
differentiating w.r.t x,we get
$\begin{array}{c}2x+\frac{2b}{a}y{y}_1+0=0\\ y_1=-\frac{ax}{by}\\ \frac{y{y}_1}{x}=\frac{a}{b}\end{array}$
again differentiating w.r.t x ,we have
$\begin{array}{c}\frac{x\left({y_1}^2+y{y}_2-y{y}_1\right)}{x^2}=0\\ {y_1}^2+y{y}_2-y{y}_1=0\\ y{y}_2+y_1^2-y{y}_1=0\end{array}$