Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.

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Start with the given polar equation: \(r^{2} + r(2\sin\theta - 6\cos\theta) = 0\).

Recall the relationships between polar and Cartesian coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r^{2} = x^{2} + y^{2}\).

Rewrite each term in the equation using these relationships: replace \(r^{2}\) with \(x^{2} + y^{2}\), replace \(r\sin\theta\) with \(y\), and replace \(r\cos\theta\) with \(x\).

Substitute these into the equation to get: \((x^{2} + y^{2}) + (2y - 6x) = 0\).

Simplify and rearrange the equation to standard Cartesian form, then analyze the resulting equation to identify the type of curve it represents (such as a circle, ellipse, parabola, or hyperbola).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar to Cartesian Coordinate Conversion

This involves translating equations from polar form (r, θ) to Cartesian form (x, y) using the relationships x = r cos θ and y = r sin θ. Understanding these conversions allows one to rewrite polar equations in terms of x and y for easier analysis.

Algebraic Manipulation of Polar Equations

Rearranging and simplifying polar equations often requires substituting r and trigonometric terms with Cartesian equivalents and then algebraically manipulating the resulting expressions. This step is crucial to express the equation in a recognizable Cartesian form.

Identification of Conic Sections

Once the equation is in Cartesian form, recognizing the type of curve (circle, ellipse, parabola, or hyperbola) involves comparing it to standard conic section equations. This helps in understanding the geometric nature of the curve represented by the original polar equation.

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