Examples of Polar Equations

[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.

r = −2 cos θ

검증된 단계별 안내

Recognize that the given equation is in polar form: \(r = -2 \cos \theta\). Here, \(r\) represents the distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.

Recall the relationship between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). These will help us understand the shape of the graph.

Multiply both sides of the equation by \(r\) to rewrite it as \(r^2 = -2r \cos \theta\). Since \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\), substitute these to get \(x^2 + y^2 = -2x\).

Rearrange the equation to standard form by bringing all terms to one side: \(x^2 + y^2 + 2x = 0\). Complete the square for the \(x\) terms to identify the conic section.

Complete the square: \(x^2 + 2x + 1 + y^2 = 1\), which simplifies to \((x + 1)^2 + y^2 = 1\). This represents a circle centered at \((-1, 0)\) with radius \(1\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for interpreting and graphing polar equations.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their radius and angle values. For equations like r = -2 cos θ, the radius can be negative, which means the point is plotted in the opposite direction of the angle θ. Recognizing how to handle negative r values is key to accurate graphing.

Conic Sections in Polar Form

Certain conic sections, such as circles, ellipses, parabolas, and hyperbolas, can be expressed in polar form. Equations like r = -2 cos θ often represent circles or other conics. Understanding how conic sections translate into polar equations helps in identifying and sketching their graphs.

Examples of Polar Equations[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.r = −2 cos θ