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# Ellipse from Geometrical Properties Problem: Figure 13-5i shows concentric circles of ISBN: 9781559533911 468

## Solution for problem 5 Chapter 13-5

Precalculus with Trigonometry: Concepts and Applications | 1st Edition

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Problem 5

Ellipse from Geometrical Properties Problem: Figure 13-5i shows concentric circles of radii 3 and 5 centered at the origin of a Cartesian coordinate system. A ray from the center at an angle of t radians to the x-axis cuts the two circles at points A and B, respectively. From point A a horizontal line is drawn, and from point B a vertical line is drawn. These two lines intersect at point P on the graph of a curve. Figure 13-5i a. On a copy of Figure 13-5i, pick other values of the angle t and plot more points using the given specifications. Connect the resulting points with a smooth curve. Does the graph seem to be an ellipse? b. Write parametric equations for the point P(x, y) in terms of the parameter t using the given geometrical description. Is the result the same as the parametric equations of an ellipse from Section 4-5? c. Plot the parametric equations on your grapher. Use equal scales on the two axes. Also, plot the two circles. Do the circles have the same relationship to the curve as in your sketch? d. The parameter t can be eliminated from the two parametric equations to give a single equation involving only the variables x and y. Clever use of the Pythagorean properties will allow you to do this. Write a Cartesian equation for this curve. How can you tell the equation represents an ellipse?

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##### ISBN: 9781559533911

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