Mathematics

This problem asks you to find a geometric property of a parametric curve by working with its normal line. The key insight is that the perpendicular distance from the origin to any line equals $\frac{|c|}{\sqrt{A^2+B^2}}$ when the line is written as $Ax + By + c = 0$. To find the normal at a point, you'll need the slope of the tangent (using $\frac{dy/d\theta}{dx/d\theta}$), then take its negative reciprocal. The elegant part: for many curves, this distance turns out to be constant—independent of $\theta$—which reveals something beautiful about the curve's geometry.

This problem asks you to find a geometric property of a parametric curve by working with its normal line. The key insight is that the perpendicular distance from the origin to any line equals $\frac{|c|}{\sqrt{A^2+B^2}}$ when the line is written as $Ax + By + c = 0$. To find the normal at a point, you'll need the slope of the tangent (using $\frac{dy/d\theta}{dx/d\theta}$), then take its negative reciprocal. The elegant part: for many curves, this distance turns out to be constant—independent of $\theta$—which reveals something beautiful about the curve's geometry.