1162. For which curve is the segment of any tangent, enclosed between the point of tangency and the x-axis, bisected by the y-axis?
Solution. The equation of the tangent at any point $(x, y)$ of the desired curve will be $Y-y=y^{\prime}(X-x)$, where $X, Y-$ are the coordinates of any point on the tangent (Chapter II, § 11).

By setting $Y=0$ in this equation, we find the abscissa $X_{0}$ of the point of intersection of the tangent with the $O x$ axis: $X_{0}=x-\frac{y}{y^{\prime}}$.

According to the condition of the problem, $X_{0}+x=0$, i.e., $2 x-\frac{y}{y^{\prime}}=0$.

Solving this differential equation of the desired curve as an equation with separable variables, we get

$$
2 \frac{d y}{y}=\frac{d x}{x} ; \quad 2 \ln |y|=\ln |x|+\ln C ; \quad y^{2}=C x .
$$

Therefore, the desired curve is a parabola with its vertex at the origin, symmetric with respect to the $O x$ axis.