# The curve amongst the family of curves represented

Question:

The curve amongst the family of curves represented by the differential equation, $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ which passes through $(1,1)$, is:

1. (1) a circle with centre on the $x$-axis.

2. (2) an ellipse with major axis along the $y$-axis.

3. (3) a circle with centre on the $y$-axis.

4. (4) a hyperbola with transverse axis along the $x$-axis.

Correct Option: 1

Solution:

$y^{2} d x-2 x y d y=x^{2} d x$

$2 x y d y-y^{2} d x=-x^{2} d x$

$d\left(x y^{2}\right)=-x^{2} d x$

$\frac{x d\left(y^{2}\right)-y^{2} d(x)}{x^{2}}=-d x$

$d\left(\frac{y^{2}}{x}\right)=-d x$

$\int d\left(\frac{y^{2}}{x}\right)=-\int d x$

$\frac{y^{2}}{x}=-x+C$ .....(1)

Since, the above curve passes through the point $(1,1)$

Then, $\frac{1^{2}}{1}=-1+C \Rightarrow C=2$

Now, the curve (1) becomes

$y^{2}=-x^{2}+2 x$

$\Rightarrow y^{2}=-(x-1)^{2}+1$

$(x-1)^{2}+y^{2}=1$

The above equation represents a circle with centre $(1,0)$ and centre lies on $x$-axis.