Which Of The Following Represents The Rectangular Equation $x^2+y^2+5x-3y=0$ In Polar Form?A. $r = -5 \sin \theta + 3 \cos \theta$B. $r = -3 \sin \theta + 5 \cos \theta$C. $r = 3 \sin \theta - 5 \cos \theta$D. $r

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Introduction

Polar coordinates provide an alternative way to represent points in a two-dimensional plane. In polar coordinates, a point is represented by its distance from the origin (r) and the angle formed with the positive x-axis (θ). Converting rectangular equations to polar form is an essential skill in mathematics, particularly in calculus and engineering applications. In this article, we will explore how to convert the rectangular equation x2+y2+5x−3y=0x^2+y^2+5x-3y=0 to polar form.

Understanding Rectangular and Polar Coordinates

Before we dive into the conversion process, let's briefly review rectangular and polar coordinates.

  • Rectangular Coordinates: In rectangular coordinates, a point is represented by its x and y coordinates. The x-coordinate represents the horizontal distance from the origin, while the y-coordinate represents the vertical distance.
  • Polar Coordinates: In polar coordinates, a point is represented by its distance from the origin (r) and the angle formed with the positive x-axis (θ).

Converting Rectangular Equations to Polar Form

To convert a rectangular equation to polar form, we need to express x and y in terms of r and θ. We can use the following relationships:

  • x=rcos⁡θx = r \cos \theta
  • y=rsin⁡θy = r \sin \theta

Substituting these expressions into the original equation, we get:

r2cos⁡2θ+r2sin⁡2θ+5rcos⁡θ−3rsin⁡θ=0r^2 \cos^2 \theta + r^2 \sin^2 \theta + 5r \cos \theta - 3r \sin \theta = 0

Simplifying the Equation

We can simplify the equation by combining like terms:

r2(cos⁡2θ+sin⁡2θ)+5rcos⁡θ−3rsin⁡θ=0r^2 (\cos^2 \theta + \sin^2 \theta) + 5r \cos \theta - 3r \sin \theta = 0

Using the trigonometric identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1, we can rewrite the equation as:

r2+5rcos⁡θ−3rsin⁡θ=0r^2 + 5r \cos \theta - 3r \sin \theta = 0

Factoring Out r

We can factor out r from the equation:

r(1+5cos⁡θ−3sin⁡θ)=0r(1 + 5 \cos \theta - 3 \sin \theta) = 0

Solving for r

To solve for r, we can divide both sides of the equation by the expression in parentheses:

r=−5cos⁡θ+3sin⁡θr = -5 \cos \theta + 3 \sin \theta

Conclusion

In this article, we have shown how to convert the rectangular equation x2+y2+5x−3y=0x^2+y^2+5x-3y=0 to polar form. By expressing x and y in terms of r and θ, we can simplify the equation and solve for r. The resulting polar equation is r=−5cos⁡θ+3sin⁡θr = -5 \cos \theta + 3 \sin \theta. This equation represents a circle centered at the origin with a radius of 5.

Answer

The correct answer is:

  • A. r=−5sin⁡θ+3cos⁡θr = -5 \sin \theta + 3 \cos \theta

Q: What is the difference between rectangular and polar coordinates?

A: Rectangular coordinates represent a point in a two-dimensional plane using its x and y coordinates. Polar coordinates, on the other hand, represent a point using its distance from the origin (r) and the angle formed with the positive x-axis (θ).

Q: How do I convert a rectangular equation to polar form?

A: To convert a rectangular equation to polar form, you need to express x and y in terms of r and θ. You can use the following relationships:

  • x=rcos⁡θx = r \cos \theta
  • y=rsin⁡θy = r \sin \theta

Substitute these expressions into the original equation and simplify.

Q: What is the correct conversion of the rectangular equation x2+y2+5x−3y=0x^2+y^2+5x-3y=0 to polar form?

A: The correct conversion is:

r=−5cos⁡θ+3sin⁡θr = -5 \cos \theta + 3 \sin \theta

Q: Why is it important to convert rectangular equations to polar form?

A: Converting rectangular equations to polar form is essential in mathematics, particularly in calculus and engineering applications. Polar coordinates provide a more intuitive and convenient way to represent points in a two-dimensional plane, making it easier to solve problems and analyze data.

Q: Can I use the same conversion process for all rectangular equations?

A: No, the conversion process may vary depending on the specific equation. You need to analyze the equation and determine the correct conversion process.

Q: What are some common applications of polar coordinates?

A: Polar coordinates have numerous applications in mathematics, physics, engineering, and computer science. Some common applications include:

  • Calculus: Polar coordinates are used to represent points in a two-dimensional plane, making it easier to solve problems and analyze data.
  • Physics: Polar coordinates are used to represent the position and velocity of objects in a two-dimensional plane.
  • Engineering: Polar coordinates are used to design and analyze systems, such as electrical circuits and mechanical systems.
  • Computer Science: Polar coordinates are used in computer graphics and game development to represent points in a two-dimensional plane.

Q: How do I practice converting rectangular equations to polar form?

A: You can practice converting rectangular equations to polar form by:

  • Solving problems: Practice solving problems that involve converting rectangular equations to polar form.
  • Using online resources: Use online resources, such as calculators and interactive tools, to practice converting rectangular equations to polar form.
  • Working with a tutor: Work with a tutor or instructor to practice converting rectangular equations to polar form.

Conclusion

Converting rectangular equations to polar form is an essential skill in mathematics, particularly in calculus and engineering applications. By understanding the differences between rectangular and polar coordinates, you can convert rectangular equations to polar form and solve problems more efficiently. Remember to practice converting rectangular equations to polar form to improve your skills and confidence.