Question

Form the differential equation of the circles represented by $y^2-2ay+x^2=a^2,$ a being arbitrary constant.

• A. $x^2(y_1^2-1)=2x^2+4xy{y}_1+2y^2{y}_1^2$
• B. $x^2(y_1^2+1)=2x^2+4xy{y}_1-2y^2{y}_1^2$
• C. $x^2(y_1^2-1)=2x^2+4xy{y}_1-2y^2{y}_1^2$
• D. $x^2(y_1^2+1)=2x^2+4xy{y}_1+2y^2{y}_1^2$

Answer 1

The correct option is D $x^2(y_1^2+1)=2x^2+4xy{y}_1+2y^2{y}_1^2$

The given equation can be written as, $x^2+{(y-a)}^2=2a^2...\left(1\right)$
Differentiating w.r.t $x$
$2x+2(y-a)y_1=0$
or $x+y{y}_1=a{y}_1..\left(2\right)$
Putting the value of a from (2) in (1), we get
or $x^2+{(y-\frac{x+y{y}_1}{y_1})}^2=2{\left(\frac{x+y{y}_1}{y_1}\right)}^2$
$x^2(y_1^2+1)=2x^2+4xy{y}_1+2y^2{y}_1^2$