Question

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.

Accepted Answer

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. 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 $$t)^2. $$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$$ \$$sin^2 t$$)$$. Group like terms involving $$\$$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. $$Use the given conditions $$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$, which is the equation of a circle with radius R$.

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 video answer for a similar problem:

Key Concepts

Parametric Equations of a Curve

Properties of the Circle Equation

Trigonometric Identities and Orthogonality Conditions

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 video answer for a similar problem:

Key Concepts

Parametric Equations of a Curve

Properties of the Circle Equation

Trigonometric Identities and Orthogonality Conditions

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.