Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. $$r=\frac{1}{2 \cos \theta+3 \sin \theta}$$

Short answer

Question: Convert the polar equation \(r = \frac{1}{2\cos\theta + 3\sin\theta}\) to a Cartesian equation and describe the resulting curve. Answer: The Cartesian equation for the given polar equation is \(y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2\). The curve represented by this equation has a complex interaction between the x and y variables, as it contains terms involving \(x^2y^2\), \(x^2\), and \(xy\). We cannot directly identify it as a specific type of curve like a circle, ellipse, or parabola.

Step-by-step solution

Find an expression for "r" in the given polar equation

Given the polar equation, $$r=\frac{1}{2 \cos \theta+3 \sin \theta}$$. We will use this equation as is for further conversion.

Replace r, x, and y using conversion formulas

We use the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). We can rewrite them as \(r = \frac{x}{\cos\theta}\) and \(\frac{y}{\sin\theta} = r\). Now, substitute the Cartesian expressions for the polar equation: $$\frac{y}{\sin \theta}=\frac{1}{2 \cos \theta+3 \sin \theta}$$.

Eliminate θ

To eliminate θ, we will use trigonometric identities. We know that \(\sin^2\theta + \cos^2\theta = 1\). Multiply both sides of the equation we got in Step 2 by \(2\cos\theta + 3\sin\theta\): $$y(\sin^2\theta+\cos^2\theta) = 2\cos\theta + 3\sin\theta$$. Now, we will express \(\sin\theta\) and \(\cos\theta\) in terms of \(x\) and \(y\). We know that $$\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}$$ and $$\sin\theta = \frac{y}{\sqrt{x^2 + y^2}}$$. Substitute these formulas into the above expression: $$y = 2\left(\frac{x}{\sqrt{x^2+y^2}}\right) + 3\left(\frac{y}{\sqrt{x^2+y^2}}\right)$$

Simplify the expression

Multiply both sides of the equation by \(\sqrt{x^2 + y^2}\) to simplify further: $$y\sqrt{x^2 + y^2} = 2x + 3y$$ Now, we can square both sides of the equation to eliminate the square root: $$(y\sqrt{x^2 + y^2})^2 = (2x + 3y)^2$$ $$y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2$$

Observe the resulting curve

Now that we have our Cartesian equation: $$y^2(x^2 + y^2) = 4x^2 + 12xy + 9y^2$$ Upon simplification, we get 3 terms containing \(x^2y^2\), \(x^2\), and \(xy\). Since there is no simple form we can reduce it to, we can't directly identify it as a specific type of curve like circle, ellipse, or parabola. However, we can observe that it is a curve that possesses both \(x\) and \(y\) terms, which suggests a complex interaction between the two variables.