Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The parametric equations \(x=a t+h\) and \(y=b t+k\) where \(a \neq 0\) and \(b \neq 0,\) represent a circle centered at \((h, k)\) when \(a=b.\)

Step-by-Step Solution

Answer

False. The given parametric equations do not represent a circle but rather a parabola. The reason is that the radius, calculated from the given equations, depends on the parameter \(t\), which is not constant as it would be required for a circle.

Step 1: Identify the general equation of a circle

A circle centered at \((h, k)\) with radius \(r\) can be represented in Cartesian coordinates as \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Compare to given parametric equations

In our case, the given parametric equations are \(x=at+h\) and \(y=bt+k\) where \(a=b\neq 0\). By substituting \(x\) and \(y\) into the circle equation, we get \((at)^2 + (bt)^2 = r^2\), which simplifies to \(a^2t^2 + b^2t^2 = r^2\). However, remember that \(a = b\), so the previous equation becomes \(2a^2t^2 = r^2\). Dividing by 2 we get \(a^2t^2 = (r^2)/2\) which implies that the radius depends on \(t\), i.e., it's different for each \(t\).

Step 3: Conclude the correctness of the statement

Because for a circle the radius is constant and doesn't depend on any parameter apart from the circle itself, the original statement is incorrect. The parametric equations given represent a different figure called a parabola, not a circle centered at (h, k).

Key Concepts

Circle Equation

Understanding the equation of a circle is fundamental in understanding the geometry around us. A circle is a collection of points that are all equidistant from a center point \((h, k)\). This equidistance is known as the radius \(r\). The standard form of a circle's equation in Cartesian coordinates is given by:
\[(x-h)^2 + (y-k)^2 = r^2\]This equation ensures that no matter the point on the circle's edge, the distance to the center \((h, k)\) remains constant at \(r\). It's crucial when analyzing circle-related problems to remember this essential feature.

Cartesian Coordinates

The Cartesian coordinate system is a grid that allows us to pinpoint any location in a plane using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

• Each point is represented as \((x, y)\).
• The center of a circle in Cartesian coordinates is \((h, k)\), representing its fixed location in the plane.

This coordinate system is handy for solving geometric problems, like determining whether a set of equations represents a circle, as it lays the framework for plotting precise shapes and paths. Understanding Cartesian coordinates helps in deeply analyzing the spatial relationships between different mathematical figures.

Equation of a Circle

The core of understanding circles lies in their equation form. Besides the standard form \((x-h)^2 + (y-k)^2 = r^2\), we often need to adapt this equation based on given data or problem statements. If you have a different set of variables or parameters, such as changes with time \(t\) or other adjustments, this standard form allows for flexibility.

• The parameters \(h\) and \(k\) translate the circle's center across the plane.
• The value of \(r\) determines the scale of the circle.

When comparing or deriving parametric equations, the 'constant radius' part is crucial, as any variation typically indicates the representation of a different shape, such as an ellipse or parabola.

Parametric Form

Parametric equations provide a method to express coordinates of points on a curve, where each coordinate is expressed in terms of another variable, commonly \(t\). For circles, a simple parametric representation is:

• \(x = r \cos(t) + h\)
• \(y = r \sin(t) + k\)

Here, \(t\) is used to trace the entire circle as it varies from 0 to \(2\pi\). This form is instrumental particularly in animations and simulations where the circular motion needs to be depicted. However, the parametric form above differs from the given ones in the exercise, revealing a misunderstanding that needs to be corrected when applying parametric equations to accurately depict circular motion or shape.