How can I find the Cartesian equation for the curve and identify whether it is a parabola, ellipse, line, circle, or limacon?

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To find the Cartesian equation for a curve defined in polar coordinates, we typically start with the polar equation given, which can involve trigonometric functions like tangent (tan) or secant (sec). Let’s break down the process step by step:

1. Understand the Polar Equation

If the given polar equation is in the form of r = f(θ), it represents a curve in the polar coordinate system. For example, if we have r = 10 tan(θ), we need to follow the next steps to convert it into Cartesian coordinates.

2. Convert Polar to Cartesian Coordinates

The relationships between polar and Cartesian coordinates are:

  • x = r cos(θ)
  • y = r sin(θ)
  • r = √(x² + y²)

For the example r = 10 tan(θ), we can use the identity tan(θ) = sin(θ)/cos(θ). Substituting this into our equation gives us:

r = 10 (sin(θ)/cos(θ))

Replace sin(θ) and cos(θ) with their Cartesian equivalents:

r = 10 (y/r)/(x/r)

Thus, we can simplify to:

r^2 = 10y/x

Multiplying both sides by x yields:

x r^2 = 10y

3. Identify the Equation

Substituting r^2 = x^2 + y^2 gives us:

x(x^2 + y^2) = 10y

At this point, you can expand and rearrange the terms to form a recognizable Cartesian equation. Depending on the resulting equation’s properties (such as degree and shape), you can classify the curve:

  • If it fits the general form of y = ax^2 + bx + c, it’s a parabola.
  • If it conforms to the forms like (x-h)²/a² + (y-k)²/b² = 1, then it’s an ellipse.
  • If it reduces to a circle’s equation, such as (x-h)² + (y-k)² = r², then it’s a circle.

Conclusion

The specific method you follow may differ depending on the original form of the polar equation. Once you have the Cartesian equation, you can easily determine the type of curve it represents based on its algebraic structure.

If you are working with a limacon, note that its equation typically involves forms like r = a ± b sin(θ) or r = a ± b cos(θ), which should also be converted similarly to identify its Cartesian form. Understanding the terminology and shapes associated with these curves will enhance both your analytical skills and engagement with the material.

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