Math Problem Statement

Rewrite the curve x2+y2=4x in polar coordinatesx^2+y^2=4x\text{ in polar coordinates}

Solution

To rewrite the curve x2+y2=4xx^2 + y^2 = 4x in polar coordinates, we can use the relationships x=rcos(θ)x = r \cos(\theta) and y=rsin(θ)y = r \sin(\theta).

Starting with the given equation:

x2+y2=4xx^2 + y^2 = 4x

Substituting the polar coordinates:

(rcos(θ))2+(rsin(θ))2=4(rcos(θ))(r \cos(\theta))^2 + (r \sin(\theta))^2 = 4(r \cos(\theta))

This simplifies to:

r2(cos2(θ)+sin2(θ))=4rcos(θ)r^2 (\cos^2(\theta) + \sin^2(\theta)) = 4r \cos(\theta)

Using the Pythagorean identity cos2(θ)+sin2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1, we have:

r2=4rcos(θ)r^2 = 4r \cos(\theta)

Dividing both sides by rr (assuming r0r \neq 0) gives:

r=4cos(θ)r = 4 \cos(\theta)

Thus, the curve in polar coordinates is:

r=4cos(θ)r = 4 \cos(\theta)

Would you like to explore more about this curve or have any questions? Here are some related questions you might find interesting:

  1. What does the graph of r=4cos(θ)r = 4 \cos(\theta) look like?
  2. How do you derive polar coordinates from Cartesian coordinates generally?
  3. What is the significance of the curve being a circle?
  4. How does this curve behave as θ\theta varies?
  5. Can you convert other conic sections to polar coordinates?

Tip: Always check the domain of θ\theta when working with polar equations!

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Math Problem Analysis

Mathematical Concepts

Polar Coordinates
Analytic Geometry

Formulas

x = r cos(θ)
y = r sin(θ)
Pythagorean identity: cos²(θ) + sin²(θ) = 1

Theorems

Conversion from Cartesian to Polar Coordinates

Suitable Grade Level

Grades 10-12