Mathematics

Mathematics

Problem

For a smooth parametric curve (x(t),y(t))(x(t),y(t)), the tangent lines at parameter values t=at=a and t=bt=b are perpendicular. State the algebraic relationship that must hold between the values of dydx\dfrac{dy}{dx} evaluated at t=at=a and t=bt=b.
Intuitive explanation

The slope of a parametric curve at any point is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}. When two lines are perpendicular, their slopes multiply to give −1-1—this is the key geometric condition. So you need to express that perpendicularity constraint using the slopes at the two parameter values.

Intuitive explanation

The slope of a parametric curve at any point is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}. When two lines are perpendicular, their slopes multiply to give −1-1—this is the key geometric condition. So you need to express that perpendicularity constraint using the slopes at the two parameter values.

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