A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.

Verified step by step guidance

1. Recall the relationships between polar and Cartesian coordinates: \(x = r \cos\theta\), \(y = r \sin\theta\), and \(r^2 = x^2 + y^2\).
2. Given the equation \(r^2 = \sec^2\theta\), rewrite \(\sec^2\theta\) as \(\frac{1}{\cos^2\theta}\), so the equation becomes \(r^2 = \frac{1}{\cos^2\theta}\).
3. Multiply both sides by \(\cos^2\theta\) to get \(r^2 \cos^2\theta = 1\).
4. Substitute \(r \cos\theta\) with \(x\) (since \(x = r \cos\theta\)), so the equation becomes \(x^2 = 1\).
5. Interpret the Cartesian equation \(x^2 = 1\) as two vertical lines \(x = 1\) and \(x = -1\), which are the Cartesian form of the given polar equation.

Key Concepts

Polar to Cartesian Coordinate Conversion

This involves expressing polar coordinates (r, θ) in terms of Cartesian coordinates (x, y) using the relationships x = r cos θ and y = r sin θ. Converting polar equations to Cartesian form helps in identifying and analyzing curves using familiar algebraic methods.

Trigonometric Identities

Trigonometric identities, such as sec²θ = 1/cos²θ and the Pythagorean identity sin²θ + cos²θ = 1, are essential for manipulating and simplifying equations involving trigonometric functions. These identities enable rewriting the given polar equation into a more workable form.

Identification of Conic Sections

Conic sections include ellipses, parabolas, and hyperbolas, each with characteristic Cartesian equations. After converting the polar equation, recognizing the standard form of these curves allows one to classify the curve correctly based on its algebraic equation.