If the angle between the radius vector and the tangent is...

If the angle between the radius vector and the tangent is censtant, then the curve is, $\begin{array}{lll}\text { (A) } r=a \cos \theta & \text { (B) } r^{2}=\alpha^{2} \cos ^{2} \theta & \text { (C) } r \text { al } 6 e^{\prime \prime}\end{array}$

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Key Concepts

Logarithmic Spiral (Equiangular Spiral)

A logarithmic spiral, or equiangular spiral, is defined by the property that the angle between the radius vector and the tangent remains constant at every point. This characteristic results in a self-similar curve whose polar equation is typically expressed in the form r = ae^(k?), where the constant k relates directly to the fixed tangent-radius angle.

Polar Coordinate System

The polar coordinate system represents points on a plane using a distance from a fixed point (the pole) and an angle from a fixed direction. This system is especially useful when dealing with curves and phenomena exhibiting radial symmetry or angular properties.

Angle Between Radius Vector and Tangent

In polar curves, the angle between the radius vector and the tangent line at a given point reveals information about the curve’s rate of change in the radial direction compared to its angular change. Setting this angle constant leads to a specific type of differential equation that characterizes the curve.

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