Problem 72 - Precalculus

Question

Equation Show that the polar equation of the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta}$$.

Step-by-Step Solution

Answer

The Cartesian equation for the hyperbola was successfully transformed into the given polar form, successfully showing that \[r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta}\], where \(e^{2}=\frac{a^{2}}{b^{2}}\].

Step 1: Express Cartesian Coordinates in Polar Form

Let's express the Cartesian coordinates in polar form. In polar coordinates, \(x=r\cos\theta\) and \(y=r\sin\theta\). So, substitute these values into the given hyperbolic equation: \[\frac{(r\cos\theta)^{2}}{a^{2}}-\frac{(r\sin\theta)^{2}}{b^{2}}=1.\] Simplify by cancelling out the \(r^{2}\) in the numerator and denominator.

Step 2: Simplify and Rearrange Equation

Arrange the equation so that everything except \(r^{2}\) is on one side and then isolate \(r^{2}\). After performing the algebraic manipulations, get: \[r^{2} =\frac{b^2}{1-(\frac{a^2}{b^2})\cos^2\theta}.\]

Step 3: Express in Required Form

Notice that in the equation \[r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta}\], the term \(e^{2}\) is equal to \(\frac{a^{2}}{b^{2}}\), the square of \(\frac{a}{b}\), which is the eccentricity of the hyperbola. Hence, the polar equation of the given hyperbola, is \[r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta}\].

Key Concepts

Hyperbola

A hyperbola is a type of conic section that appears when a plane cuts through a double cone in such a way that the angle of the plane is less than the angle of the cone’s side. It consists of two separate curves, called branches, which mirror each other. A distinctive feature of hyperbolas is their two foci, or fixed points. The graph of a hyperbola can look like two opposite arches, and it resembles an hourglass figure when rotated at certain angles.

In terms of equations, a hyperbola in the Cartesian plane is generally expressed as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) represent the distances from the center of the hyperbola to the vertices and to the co-vertices, respectively. What makes hyperbolas intriguing is their open structure, which means they do not form a closed curve but continue infinitely in two opposite directions.

Cartesian Coordinates

In mathematics, Cartesian coordinates are a system that specifies each point uniquely in a plane by a pair of numerical coordinates. These coordinates are signed distances to the point from two fixed perpendicular oriented lines. Typically, these are denoted as \( (x, y) \) in two-dimensional spaces, where \(x\) represents the horizontal position and \(y\) the vertical position. This system allows for a flat, grid-like representation of points, akin to using a map.

When relating this to other coordinate systems like polar coordinates, transformations can be applied. This is done using the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

These equations express Cartesian coordinates in terms of polar coordinates \((r, \theta)\), making it easier to switch between the two systems. This transformation is especially useful when dealing with equations of conic sections such as hyperbolas.

Eccentricity

Eccentricity is a crucial concept in understanding the shape of any conic section. It measures the deviation of the conic from being a perfect circle. The value of eccentricity, denoted by \(e\), helps to classify conics:

• If \(e = 0\), the conic is a circle.
• If \(0 < e < 1\), it's an ellipse.
• If \(e = 1\), it forms a parabola.
• If \(e > 1\), it's a hyperbola.

For hyperbolas, the eccentricity is always greater than 1, indicating its nature as an open, non-closed curve. In equations, the eccentricity can relate lengths involved in the formula, such as \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus. For the discussed transformation into polar form, the eccentricity appears as \(e^2 = \frac{a^2}{b^2}\), iterating the link between geometry and algebra.

Polar Form

The polar form is another way to express mathematical equations, especially when dealing with curves centered around a point, rather than along axes. In the polar coordinate system, each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The polar form is well-suited for conic sections, like circles, ellipses, hyperbolas, and parabolas, allowing for more intuitive representation, especially in equations involving rotations or infinite extents like hyperbolas.

To express equations in polar coordinates, transformations exploit formulas like:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

For instance, converting the Cartesian equation of a hyperbola to its polar form involves substituting \(x\) and \(y\) in terms of \(r\) and \(\theta\). The resulting expression, like \(r^2 = \frac{-b^2}{1-e^2 \cos^2 \theta}\), shows how polar equations manage symmetry and periodicity differently from Cartesian representations, often simplifying analysis and solution of complex geometric problems.