Question

For the following exercises, find the polar equation for the curve given as a Cartesian equation. $$ y^{2}=4+x^{2} $$

Short answer

The polar equation is \( r^2 = -\frac{4}{\cos 2\theta} \).

Step-by-step solution

Understand the Cartesian Equation

The given Cartesian equation is \( y^2 = 4 + x^2 \). This equation represents a curve in the Cartesian coordinate system and our task is to convert it into a polar equation.

Convert Cartesian Coordinates to Polar Coordinates

Recall that the polar coordinates \((r, \theta)\) are related to the Cartesian coordinates \((x, y)\) by the equations: \( x = r \cos \theta \) and \( y = r \sin \theta \). We will use these relationships to express the Cartesian equation in terms of \( r \) and \( \theta \).

Express x and y in Polar Form

Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation: \( (r \sin \theta)^2 = 4 + (r \cos \theta)^2 \).

Simplify the Equation

Simplify the equation to get:\( r^2 \sin^2 \theta = 4 + r^2 \cos^2 \theta \).

Isolate the Terms Involving r

Rearrange to write all terms involving \( r^2 \) on one side:\( r^2 \sin^2 \theta - r^2 \cos^2 \theta = 4 \).

Factor Out r^2 on the Left Side

Factor \( r^2 \) from the left side: \( r^2 (\sin^2 \theta - \cos^2 \theta) = 4 \).

Use Trigonometric Identity

Recognize that \( \sin^2 \theta - \cos^2 \theta = -\cos 2\theta \). Substitute into the equation to get:\( r^2 (-\cos 2\theta) = 4 \).

Solve for r^2

Rewrite the equation to solve for \( r^2 \): \( r^2 = \frac{4}{-\cos 2\theta} \) or equivalently, \( r^2 = -\frac{4}{\cos 2\theta} \).

Write the Solution as a Polar Equation

The polar equation for the given Cartesian equation is \( r^2 = -\frac{4}{\cos 2\theta} \).

Key concepts

Cartesian Coordinates

A Cartesian coordinate system is a two-dimensional plane where each point is uniquely determined by a pair of numerical coordinates. These coordinates, usually labeled as \((x, y)\), represent a point's horizontal and vertical distances from the origin. The origin is typically denoted as \((0, 0)\). This system is widely used in mathematics to graphically represent equations and data.
In this system, the horizontal axis is called the x-axis, and the vertical axis is the y-axis. Points are described as their coordinates \((x, y)\). The Cartesian plane is excellent for visualizing simple algebraic equations, like lines and curves. The equation provided, \(y^2 = 4 + x^2\), is expressed in Cartesian form, depicting a specific curve. Our goal is to convert this form into polar coordinates, a different type commonly used for circular patterns and rotations.

Coordinate Transformation

Coordinate transformation involves changing the coordinates of a point from one coordinate system to another. In our case, we're transitioning from Cartesian coordinates to polar coordinates. This transformation process entails using specific relationships between the two systems.

• In Cartesian coordinates, points are denoted as \((x, y)\).
• In polar coordinates, points are represented by a radius \(r\) and an angle \(\theta\).

The equations that facilitate this transformation are:

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

These equations allow the transition between the systems, making it easier to express the given equation \(y^2 = 4 + x^2\) in terms of \(r\) and \(\theta\). By substituting these relationships into the Cartesian equation, we achieve a new form that captures the same geometry but in another coordinate system.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. To simplify complex equations, we use these identities, allowing us to solve for unknowns efficiently.For the equation transformation, we apply the identity: \( \sin^2 \theta - \cos^2 \theta = -\cos 2\theta \)This identity emerges useful here to simplify the transformed equation into a manageable form. Applying this identity, we re-write and solve our equation:

• Substitute: \(r^2 (\sin^2 \theta - \cos^2 \theta) = 4\).
• Re-write: \(r^2 (-\cos 2\theta) = 4\).

By recognizing and using this identity, we turned a complex mixture of sines and cosines into a simpler expression. This simplification process clarifies the relationship between polar coordinates \(r^2\) and angle \(\theta\) in our final polar equation \(r^2 = -\frac{4}{\cos 2\theta}\).