91Ó°ÊÓ

To convert the given polar equation $$r=\frac{1}{2-2 \sin \theta}$$ to Cartesian coordinates, we use the following equations: $$x = r \cos \theta \; \text{and} \; y = r \sin \theta$$ Substitute the given polar equation for r into both equations: $$x = \frac{\cos \theta}{2-2 \sin \theta}$$ $$y = \frac{\sin \theta}{2-2 \sin \theta}$$ We need to eliminate theta to get the equation in the Cartesian coordinate system. Multiply the second equation by 2 and add the first one to find the value of x: $$x = 2y + 1$$ Now, we will substitute the value of x in the second equation: $$y = \frac{\sin \theta}{2-2 \sin \theta}$$ $$y = \frac{\sin \theta}{2(1 - \sin\theta)}$$ $$y(1 - \sin\theta) = \sin \theta$$ $$y - y \sin\theta = \sin \theta$$ $$y = \sin \theta (\frac{1}{1-y})$$ Now, we use the sine angle addition formula: $$\sin(\theta + \frac{\pi}{2}) = y$$ This looks like the equation of a parabola in Cartesian coordinates.

The equation is: $$x = 2y + 1$$ Subtract 2y and 1 from both sides to make it the standard form: $$x-2y-1=0$$ So the equation in Cartesian coordinates is: $$x - 2y - 1 = 0$$ This is the equation of a parabola with a vertical axis and vertex at (1, 0). The parabola opens horizontally to the right due to the coefficient of y being positive.

For the parabola $$x - 2y - 1 = 0$$, we have: Vertex (V): (1, 0) Focus (F): A parabola has a single focus, which can be found by adding 1/4 to the x-coordinate of the vertex. In this case, the focus will be at (5/4, 0). Directrix (D): A parabola has a single directrix, which is a vertical line whose x-coordinate can be found by subtracting 1/4 from the x-coordinate of the vertex. So the equation of the directrix is $$x = 3/4$$. Asymptotes: Parabolas do not have asymptotes.

1. Plot the vertex V at (1,0). 2. Plot the focus F at (5/4, 0). 3. Draw the vertical directrix line at $$x = 3/4$$. 4. Sketch the parabola with its axis parallel to the x-axis and opening to the right, as indicated by the positive coefficient of y. 5. The graph should now resemble a parabola with the given important points.

To ensure that the graph is correct, use a graphing utility to plot the polar equation $$r=\frac{1}{2-2 \sin \theta}$$ and compare it to the Cartesian graph obtained in Step 4. The graphs should match, confirming that the graph is correct.