If $f(x)$ and $g(x)$ are two solutions of differential equation, then find $\frac{f'(x_1)}{g'(x_2)}$

Question

Let $c_1$ and $c_2$ be two integral curves of differential equation $\frac{dy}{dx}=\frac{x^2-y^2}{x^2+y^2}$. A line passing through origin meets $c_1$ at $P(x_1,y_1)$ and $c_2$ at $Q(x_2,y_2)$. If $c_1:y=f(x)$ and $c_2:y=g(x)$, then find the value of $\frac{f'(x_1)}{g'(x_2)}$?

How should I proceed after finding solution of differential equation by putting $y=xt$? Given answer is $1$. Is there any valid reasoning to say that answer would be $1$ without actually solving differential equation?

Answer 1

Hint: The line through the origin is $y=tx$, where $t\in \mathbb R$ is slope. Then

$$f'(x_1)=\frac{x_1^2-(tx_1)^2}{x_1^2+(tx_1)^2}$$

$$g'(x_2)=\frac{x_2^2-(tx_2)^2}{x_2^2+(tx_2)^2}$$

Then $\frac{f'(x_1)}{g'(x_2)}$=...