This solution will guide you through finding the curve based on the given geometric condition. We will use differential equations to relate the curve's properties to the condition described.

Key Concept: Equation of the Normal to a Curve

Let $P(x_0, y_0)$ be a point on a curve.

1. Slope of the Tangent: The slope of the tangent to the curve at $P(x_0, y_0)$ is given by $\frac{dy}{dx}$ evaluated at $(x_0, y_0)$. We often denote this as $m_t = \frac{dy}{dx}$.
2. Slope of the Normal: The normal to the curve at $P(x_0, y