Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
Let $C$ be a curve such that the normal at any point $P$ on it meets $x-$axis $y-$axis at $A$ and $Y$ respectively. If $BP : PA = 1:2$ (internally) and the curve passes through the point $(0,4)$ then which of the following alternative(s) is/are correct?
The curves passes through $(\sqrt{10},-6)$
The equation of tangent at $(4,4\sqrt{3})$ is $2x+\sqrt{3}y = 20$
The differential equation for the curve is $yy'+2x = 0$
The curve represent a hyperbola
Step-by-Step Solution
Key Concept: Normal equations and geometric constraints on intercepts lead to separable differential equations that integrate directly.
The equation of the normal at point $P(x,y)$ is $(Y-y) = -\frac{dx}{dy}(X-x)$. The normal intersects the x-axis at $A = (x+y\frac{dx}{dy}, 0)$. From the geometric condition that the length $OA$ equals twice the x-coordinate of $P$, we get $x + y\frac{dx}{dy} = 2x$, so $y\frac{dx}{dy} = x$. This gives $\frac{dy}{dx} = \frac{2x}{y}$ (from rearrangement). Integrating: $\int y\,dy = \int 2x\,dx$ yields $\frac{y^2}{2} = x^2 + C$. Using the point $(4,0)$: $C = 8$, so $y^2 = 2x^2 + 16$ (a hyperbola).
Correct Answer: 1,4