Question

Show that the polar equation $$r^{2}-2 r(a \cos \theta+b \sin \theta)=R^{2}-a^{2}-b^{2}$$ describes a circle of radius \(R\) centered at \((a, b)\)

Short answer

Question: Show that the polar equation \(r^2 - 2r(a \cos\theta + b \sin\theta) = R^2 - a^2 - b^2\) represents a circle with center \((a, b)\) and radius \(R\). Answer: The polar equation can be converted into the rectangular equation \((x - a)^2 + (y - b)^2 = R^2\) which is the standard form of a circle with center \((a, b)\) and radius \(R\). Therefore, the given polar equation represents a circle of radius \(R\) centered at \((a, b)\).

Step-by-step solution

Convert the polar equation to rectangular form

To convert the given polar equation into rectangular form, use the following conversions: $$x = r\cos\theta$$ and $$y = r\sin\theta$$ We also know that $$r^2 = x^2 + y^2$$ Thus, the given polar equation can be rewritten as: $$(x^2 + y^2) - 2r(a \cos\theta + b \sin\theta) = R^2 - a^2 - b^2$$

Rewrite the equation in rectangular form

First, substitute the conversions we found in step 1 in the equation: $$x^2 + y^2 - 2a(x\cos\theta) - 2b(y\sin\theta) = R^2 - a^2 - b^2$$ Remember that \(x = r\cos\theta\) and \(y = r\sin\theta\), so $$x^2 + y^2 - 2ax - 2by = R^2 - a^2 - b^2$$

Rewrite the equation in standard form

Rearrange the terms and complete the square on the left side of the equation to obtain the standard form of a circle equation: $$\begin{aligned} (x^2 - 2ax) + (y^2 - 2by) + a^2 + b^2 &= R^2 \\ (x - a)^2 + (y - b)^2 &= R^2 \end{aligned}$$ This is the standard form of a circle with center \((a, b)\) and radius \(R\). Therefore, the given polar equation does indeed describe a circle of radius \(R\) centered at \((a, b)\).

Key concepts

Converting Polar to Rectangular Coordinates

Understanding how to convert polar to rectangular coordinates is essential when dealing with equations that involve positions and angles. In the polar coordinate system, any point in a plane is represented by a distance from a reference point and an angle from a reference direction. The reference point is usually called the origin, denoted as the pole, and the ray from the origin in the reference direction is the polar axis.

The relationship between polar \( (r, \theta) \) and rectangular \( (x, y) \) coordinates can be expressed with two simple equations: \[ x = r\cos(\theta) \] and \[ y = r\sin(\theta) \]. These relationships arise from the definition of the trigonometric functions cosine and sine.

To convert a polar equation to rectangular coordinates, we use these substitutions. For example, \[ r^2 = x^2 + y^2 \] is derived from the Pythagorean theorem applied to the right triangle formed by the point (x, y), the x-axis, and a line perpendicular to the x-axis that passes through the point.

Standard Form of a Circle

In coordinate geometry, the standard form of a circle's equation provides a clear representation of the circle's properties. The standard form is expressed as \[ (x - h)^2 + (y - k)^2 = R^2 \], where \( (h, k) \) denotes the center of the circle, and \( R \) is the circle's radius.

When a quadratic equation in two variables is given, such as in our exercise, converting it to this standard form allows us to easily identify the circle's center and radius. This process usually involves techniques such as 'completing the square,' which we will touch upon next. Recognizing the standard form helps in visualizing the circle within the Cartesian coordinate system and is a fundamental aspect of analyzing geometric shapes algebraically.

Completing the Square

Completing the square is a method used to solve quadratic equations and also to convert the general form of a circle's equation into its standard form. By adding and subtracting appropriate terms, we can create a perfect square trinomial from a quadratic expression. A perfect square trinomial is an expression that can be factored into a binomial squared, i.e., \[ (x - h)^2 \] or \[ (y - k)^2 \].

Here's how completing the square works in a general sense: Let's take a quadratic expression \[ ax^2 + bx + c \], we start by isolating the variable terms \[ ax^2 + bx \], then divide by the coefficient of \( x^2 \) if it's not 1, which makes it easier to handle. Next, we add and subtract \( (\frac{b}{2a})^2 \) inside the brackets, and finally, group the perfect square trinomial and the constant terms separately.

In our exercise example, to convert the equation \[ x^2 - 2ax + y^2 - 2by \] to standard form, we added \( a^2 \) and \( b^2 \) to both sides to complete the squares for \( x \) and \( y \) respectively.

Polar and Rectangular Coordinates Relationship

The relationship between polar and rectangular coordinates is fundamental in connecting geometry with trigonometry. Both coordinate systems have their unique purposes and applications, and understanding their relationship enriches the comprehension of plane geometry and can simplify calculations.

The conversion formulas \[ x = r\cos(\theta) \] and \[ y = r\sin(\theta) \] build this bridge, enabling us to switch between the polar coordinates which are natural for circular and radial contexts, and rectangular coordinates which are conducive for Cartesian grids and algebraic operations. Shapes that may be complex to express in one system might be simpler in the other. For instance, a circle centered at the pole is easily represented in polar coordinates, whereas circles centered elsewhere are more straightforwardly dealt with in rectangular coordinates after conversion.

In real-world applications, such as navigation and robotics, polar coordinates are often used due to the rotational aspects of these fields, while rectangular coordinates are predominant in computer graphics and layout design.