Question

Let $f_1,f_2$ and $f_3$ be three curves satisfying the differential equation $\left(1-y^2\right)dx=2xydy$ and pass through $(0,2)$. If $f_3$ cuts the curves $f_1$ and $f_2$ at $A$ and $B$ respectively and one of the curve is passing through $C(5,-1)$, then find the area of $△ABC$.

Answer 1

Answer: 5

$\Theta$ Given differential equation can be written as
$\int \frac{dx}{x}=\int \frac{2y}{1-y^2}dy=\int \frac{2y}{y^2-1}dy$
$\Rightarrow \ln x=-\ln \left(y^2-1\right)+\ln c$
$\Rightarrow x\left(y^2-1\right)=c$
$\Theta (0,2)\text{ is on it }\Rightarrow c=0$
$\therefore x\left(y^2-1\right)=0\Rightarrow x=0,y=\pm 1$
Area of $△ABC=\frac{1}{2}\times 2\times 5=5$