General equations for a circle Prove that the equations  
X = a cos t + b sin t,  y = c cos t + d sin t  
where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

Verified step by step guidance

1. Start with the given parametric equations of the curve: \(X = a \cos t + b \sin t\) and \(y = c \cos t + d \sin t\), where \(a, b, c,\) and \(d\) are real numbers.
2. Square both equations and add them together to find an expression for \(X^2 + y^2\): \(X^2 + y^2 = (a \cos t + b \sin t)^2 + (c \cos t + d \sin t)^2\).
3. Expand the squares using the distributive property: \(X^2 + y^2 = (a^2 \cos^2 t + 2ab \cos t \sin t + b^2 \sin^2 t) + (c^2 \cos^2 t + 2cd \cos t \sin t + d^2 \sin^2 t)\).
4. Group like terms involving \(\cos^2 t\), \(\sin^2 t\), and \(\cos t \sin t\): \(X^2 + y^2 = (a^2 + c^2) \cos^2 t + (b^2 + d^2) \sin^2 t + 2(ab + cd) \cos t \sin t\).
5. Use the given conditions \(a^2 + c^2 = b^2 + d^2 = R^2\) and \(ab + cd = 0\) to simplify the expression: \(X^2 + y^2 = R^2 \cos^2 t + R^2 \sin^2 t + 0 = R^2 (\cos^2 t + \sin^2 t)\). Since \(\cos^2 t + \sin^2 t = 1\), this reduces to \(X^2 + y^2 = R^2\), which is the equation of a circle with radius \(R\).

Key Concepts

Parametric Equations of a Curve

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. In this problem, x and y are given in terms of trigonometric functions of t, which allows describing complex curves like circles by varying t over an interval.

Properties of the Circle Equation

A circle with radius R centered at the origin satisfies x² + y² = R². To prove a parametric form represents a circle, one must show that substituting x(t) and y(t) into this equation yields a constant radius R, independent of t.

Trigonometric Identities and Orthogonality Conditions

The conditions a² + c² = b² + d² = R² and ab + cd = 0 ensure orthogonality and equal magnitude of vectors formed by coefficients. These constraints use trigonometric identities to guarantee that the parametric equations trace a circle rather than an ellipse or other shape.