Question

Consider the parametric 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. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

Short answer

In summary, the given parametric equations represent an ellipse in the form \(Ax^2 + Bxy + Cy^2 = K\). If \(ab + cd = 0\), the ellipse's axes are aligned with the x and y-axes, and if \(ab + cd = 0\) and \(c^2 + d^2 = a^2 + b^2 \neq 0\), the parametric equations represent a circle.

Step-by-step solution

Replace x and y with the given parametric equations

We are given: $$x = a \cos t + b \sin t \quad \text{and} \quad y = c \cos t + d \sin t$$ We will use these expressions to eliminate the parameter \(t\) and obtain an equation relating \(x\) and \(y\).

Use identities to eliminate t

To eliminate \(t\), first square both x and y equations: $$(1) \quad x^2 = (a \cos t + b \sin t)^2$$ $$(2) \quad y^2 = (c \cos t + d \sin t)^2$$ We will use the trigonometric identity \(\cos^2 t + \sin^2 t = 1\) to eliminate \(t\) from equations (1) and (2). To do this, we add equations (1) and (2) which gives: $$x^2 + y^2 = (a^2 + c^2) \cos^2 t + (b^2 + d^2) \sin^2 t + 2(ab + cd) \cos t \sin t$$ Now, let's use the identity \(\sin^2 t = 1 - \cos^2 t\). We rewrite the equation: $$x^2 + y^2 = (a^2 + c^2 - b^2 - d^2) \cos^2 t + (b^2 + d^2) + 2(ab + cd) \cos t \sin t$$ Next, we divide by \(x^2 + y^2\): $$1 = \left(\frac{a^2 + c^2 - b^2 - d^2}{x^2 + y^2}\right) \cos^2 t + \frac{b^2 + d^2}{x^2 + y^2} + 2\left(\frac{ab + cd}{x^2 + y^2}\right) \cos t \sin t$$ Now, we write this in the form of a quadratic equation: $$A \cos^2 t + B \cos t \sin t + C = 0$$ where \(A = \frac{a^2 + c^2 - b^2 - d^2}{x^2 + y^2}, B = 2\left(\frac{ab + cd}{x^2 + y^2}\right),\) and \(C = \frac{b^2 + d^2}{x^2 + y^2} - 1\). Now, \(\cos^2 t\) and \(B \cos t \sin t\) must be replaced with expressions involving \(x\) and \(y\). We can do this by rewriting the parametric equations as: $$(3) \quad \cos t = \frac{x - b\sin t}{a}$$ $$(4) \quad \sin t = \frac{y - c\cos t}{d}$$ We will use equations (3) and (4) to replace \(\cos t\) and \(\sin t\) in the quadratic equation.

Replace cos t and sin t with expressions involving x and y

We square equations (3) and (4) and add them: $$\cos^2 t + \sin^2 t = \left(\frac{x - b\sin t}{a}\right)^2 + \left(\frac{y - c\cos t}{d}\right)^2 = 1$$ This equation can be rearranged to the general form of an ellipse: $$Ax^2 + Bxy + Cy^2 = K$$ Now we can answer parts (a), (b), and (c).

Part a: Ellipse of the form \(Ax^2+Bxy+Cy^2=K\)

The equation we derived: $$Ax^2 + Bxy + Cy^2 = K$$ Represents an ellipse with the given parametric equations, as required.

Part b: Axes aligned with x and y axes

For the axes of the ellipse to be aligned with the x and y-axes, the \(xy\) term should not be present. This means \(B = 2\left(\frac{ab + cd}{x^2 + y^2}\right) = 0\). Therefore, it is required that \(ab + cd = 0\).

Part c: Representing a circle

In order for the ellipse to be a circle, the coefficients of the \(x^2\) and \(y^2\) terms must be equal. Using the values of \(A, B, C\), we need: $$\frac{a^2 + c^2 - b^2 - d^2}{x^2 + y^2} = \frac{b^2 + d^2}{x^2 + y^2}$$ Rearranging, we get: $$c^2 + d^2 = a^2 + b^2 \neq 0$$ Thus, if \(ab + cd = 0\) and \(c^2 + d^2 = a^2 + b^2 \neq 0\), the parametric equations represent a circle, as required.

Key concepts

Parametric Equations

Parametric equations define a group of quantities as functions of one or more independent variables called parameters. In the context of 2D geometry, they provide a method to describe a curve by defining the coordinates of points on the curve as functions of a single parameter. For instance, the parametric equations
\[ x = a \cos t + b \sin t, \ y = c \cos t + d \sin t \]
involve the parameter \(t\), which typically represents an angle. Parametric equations are particularly useful because they can describe a wider range of curves than standard Cartesian equations, including lines, circles, ellipses, and more complex shapes. They can also simplify certain calculations, such as finding tangents or computing arc lengths.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Based on the angle of intersection, the resulting curve can be a circle, an ellipse, a parabola, or a hyperbola. When the parametric equations describe conic sections like in our problem, they can be manipulated to reveal the classic conic sections' Cartesian forms, such as
\[ A x^{2}+B x y+C y^{2}=K \]
for an ellipse. By squaring and manipulating the parametric equations, the Cartesian form emphasizes the shape's general features rather than the point-by-point description given by parametrics.

Ellipse Equation

The equation of an ellipse in Cartesian coordinates typically has the form
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
with the center located at (h, k) and the lengths of the major and minor axes being 2a and 2b, respectively. Given the parametric equations, an ellipse equation is derived by eliminating the parameter, showing its dependence on \(x\) and \(y\) only. This can result in a more general form, as we find in the provided exercise, where
\[ A x^{2}+B x y+C y^{2}=K \]
symbolizes an ellipse with axes not necessarily aligned with the coordinate axes. When \(ab+cd=0\), the absence of an \(xy\) term reveals that the axes are aligned with the coordinate axes, leading to the traditional ellipse formula.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They play a crucial role in simplifying expressions involving trigonometric functions, as shown in the step-by-step solution of our problem. The identity
\[ \cos^2 t + \sin^2 t = 1 \]
is fundamental in trigonometry and is used to transform the sum of squares of sine and cosine into a single entity. These identities allow us to manipulate and eliminate the parameter when converting parametric equations to their corresponding Cartesian equations, facilitating the transformation of the given parametric equations into the standard form of an ellipse or a circle. Understanding these identities and their applications is essential for solving problems involving trigonometric functions.