Question

Prove that the equations $$x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t$$ where \(a, b, c,\) and \(d\) are real numbers, describe a circle of radius \(R\) provided \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0\)

Short answer

Question: Show that the parametric equations \(x=a \cos{t}+b\sin{t}\) and \(y=c\cos{t}+d\sin{t}\) describe a circle with radius R, given the conditions \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0\). Answer: The given parametric equations describe a circle of radius \(R\) with its center at the origin (0, 0), as shown by the equation \({(x-0)}^2+{(y-0)}^2=R^{2}\).

Step-by-step solution

Write the parametric equations for x and y

The given parametric equations are $$x=a \cos{t}+b\sin{t}, \quad y=c\cos{t}+d\sin{t}$$

Isolate cosine and sine terms

Separate the cosines and sines in both equations: $$x=a\cos{t}+b\sin{t}\Longrightarrow\cos{t}=\frac{x-b\sin{t}}{a}$$ $$y=c\cos{t}+d\sin(t)\Longrightarrow\sin{t}=\frac{y-c\cos{t}}{d}$$

Square the equations and sum them together

Square both equations and add them together: $$\left(\frac{x-b\sin{t}}{a}\right)^2 + \left(\frac{y-c\cos{t}}{d}\right)^2 = \cos^2{t} + \sin^2{t}$$

Simplify the equation

Expand and simplify the summation to obtain a single equation: $$\frac{{(x-b\sin{t})}^2}{a^2} + \frac{{(y-c\cos{t})}^2}{d^2} = 1$$

Utilize the given conditions

Incorporate the given conditions, \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0\), into the equation: $$\frac{{(x-b\sin{t})}^2+b^{2}\sin^2{t}}{{(a^{2}+c^{2})}{(1-\cos^2{t})}} + \frac{{(y-c\cos{t})}^2+c^{2}\cos^2{t}}{(b^{2}+d^{2})(\cos^2{t})} = 1$$ Since \({1-\cos^2{t}}=\sin^2{t}\) and \({(a^{2}+c^{2})}{(1-\cos^2{t})}=(b^{2}+d^{2})(\cos^2{t})\), we can simplify the equation further: $$\frac{{(x-b\sin{t})}^2+b^{2}\sin^2{t}}{(b^{2}+d^{2})(\cos^2{t})} +\frac{{(y-c\cos{t})}^2+c^{2}\cos^2{t}}{{(a^{2}+c^{2})}{(1-\cos^2{t})}} = 1$$

Complete the square for x and y terms

Combine the terms and complete the square for x and y separately: $${(x-b\sin{t})}^2+b^{2}\sin^2{t} + {(y-c\cos{t})}^2+c^{2}\cos^2{t}= {(a^{2}+c^{2})}{(1-\cos^2{t})}={(b^{2}+d^{2})}{\cos^2{t}}$$

Eliminate trigonometric functions

Simplify the equation and eliminate trigonometric functions: $${(x-b\sin{t})}^2+b^{2}\sin^2{t}-c^{2}\cos^2{t}={(y-c\cos{t})}^2$$

Rearrange the equation

Rearrange the equation in the form of a standard circle equation \({(x-h)}^2+{(y-k)}^2=R^{2}\): $$(x-0)^2 +(y-0)^2 = (a^2+c^2)$$ Finally, we find that the given parametric equations describe a circle of radius \(R\), as \({(x-0)}^2+{(y-0)}^2=R^{2}\). The center of the circle is at the origin (0, 0), and the radius is equal to \(R\).

Key concepts

Circle Equation

The equation of a circle can be expressed mathematically in various forms. The standard form of a circle's equation with its center at the origin is \[(x-h)^2 + (y-k)^2 = R^2\] where

• \( (h, k) \) is the center of the circle, and for a circle centered at the origin, \( h = 0 \) and \( k = 0 \).
• \( R \) is the radius of the circle.

When working with parametric equations as shown, \( x \) and \( y \) are expressed in terms of another variable (usually \( t \), which represents time or an angle).This can be used to describe the position along the circle as the parameter changes.Expressing \( x \) and \( y \) in terms of \( \cos t \) and \( \sin t \) ensures they move in a circular path due to the periodic nature of these functions.The conditions provided \( a^{2}+c^{2}=b^{2}+d^{2}=R^{2} \) ensure that the parametric representation conforms to that of a circle.These conditions ensure that the circle has equal radii in both the horizontal and vertical directions, maintaining the circular shape instead of an ellipse, guaranteeing symmetry about the axes.

Trigonometric Functions

Trigonometric functions, specifically \( \cos \) and \( \sin \), play a critical role in forming the foundation of parametric equations describing circular motion. These functions are based on the unit circle, a circle with a radius of one centered at the origin.

• The function \( \cos(t) \) corresponds to the x-coordinate of the point on the unit circle.
• The function \( \sin(t) \) corresponds to the y-coordinate.

Through their periodic properties, \( \cos \) and \( \sin \) repeat their values every \( 2\pi \) radians (360 degrees), allowing them to trace out circular paths over time.For the given parametric equations, the coefficients \( a, b, c, \) and \( d \) modulate the input angle \( t \) to scale and position the circle. Overall, these functions provide the oscillatory nature required to maintain circular motion, evidenced by the discovery that \( \cos^2(t) + \sin^2(t) = 1 \), a constant regardless of the angle \( t \).

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This operation is useful for revealing certain geometric properties, such as identifying the center and radius of a circle from its equation. To "complete the square" for a term like \( x^2 + px \), you rearrange it to \[(x + \frac{p}{2})^2 - (\frac{p}{2})^2\]which simplifies the quadratic expression into a form that highlights a transformation. In the context of circles defined by parametric equations:

• Completing the square allows isolating terms to resemble the standard circle equation.
• It reveals the squared terms that directly relate to the radius and potential offsets for the circle's center.

This process, seen when adjusting the expressions for \( x \) and \( y \), is pivotal in confirming those equations represent a circle, including ensuring all terms involving \( \sin t \) and \( \cos t \) are correctly balanced.

Geometric Interpretation

The geometric interpretation of the given parametric equations is that they describe a circle in the xy-plane.

• When the conditions \( a^2 + c^2 = b^2 + d^2 = R^2 \) and \( ab + cd = 0 \) are met, they ensure an equal contribution from each direction \( x \) and \( y \) in forming the circular path.
• These conditions eliminate any potential skewing that might arise if one component dominated, thus maintaining a regular circular shape rather than an ellipse.

In essence, each parameter in the equations involves the trigonometric identity \( \cos^2(t) + \sin^2(t) = 1 \), keeping the distance constant for all angles \( t \).The expressions make sure any point \( (x, y) \) they generate lies exactly \( R \) units from the origin, echoing the definition of a circle.Through careful manipulation and satisfaction of these conditions, a mathematical circle with its radius centered at the coordinate origin is illustrated with elegance.