A normal is drawn at a point P(x, y) of a curve. It meets...

A normal is drawn at a point $P(x, y)$ of a curve. It meets thex-axis at $Q$. If $P Q$ is of constant length $k$, then show that the differential equation describing such curves is $y \frac{d y}{d x}=\pm \sqrt{k^{2}-y^{2}}$ Find the equation of such a curve passing through $(0, k)$.

Key Concepts

Integration Techniques

Once the differential equation is separated, solving it generally requires integration methods. When the integrals involve expressions like square roots of quadratic polynomials, trigonometric substitutions often provide a viable method to evaluate them.

Initial Value Problems

The process of determining a specific solution from a general solution involves applying initial conditions. This allows for the constant of integration to be identified, ensuring that the solution curve passes through a given point.

Separable Differential Equations

A separable differential equation is one in which the variables can be separated on opposite sides of the equation, permitting integration with respect to each variable individually. This technique is fundamental for solving the derived equation from the geometric constraint.

Differential Equation Formation

Translating a geometric condition into a mathematical language often involves formulating a differential equation. In this context, using the properties of the normal line and the fixed distance creates an equation that relates the coordinates and the derivative of the function.

Normal to a Curve

The normal line at a point on a curve is the line that is perpendicular to the tangent at that point. Its slope is the negative reciprocal of the derivative of the curve, allowing geometric conditions (such as location where the normal intercepts an axis) to be expressed in terms of the derivative.

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