Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
Through any point $(x, y)$ of a curve which passes through the origin, lines are drawn parallel to the co-ordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of:
circles
pair of straight lines
parabolas
rectangular hyperbolas
Step-by-Step Solution
Key Concept: The relationship between areas $POM$ and $PON$ translates to an integral condition that becomes a differential equation upon differentiation.
Given that $P(x,y)$ lies on a curve through origin with $PN$ and $PM$ parallel to axes, the area $POM = \int_0^x y\,dx$ and $PON = xy - \int_0^x y\,dx$. If $2(POM) = PON$, then $3\int_0^x y\,dx = xy$. Differentiating gives $3y = x\frac{dy}{dx} + y$, so $2y = x\frac{dy}{dx}$. Solving this separable equation: $\frac{dy}{y} = 2\frac{dx}{x}$ yields $\log y = 2\log x + C$, giving $y = Cx^2$, which is a parabola.
Correct Answer: 3