Question

Find a polar equation for each conic section. Assume one focus is at the origin.

Short answer

Answer: The polar equations for the conic sections with one focus at the origin are as follows: Ellipse: \(r^2 = \frac{a^2b^2}{a^2 \sin^2{\theta} + b^2 \cos^2{\theta}}\) Parabola: \(r = \frac{4a\cos{\theta}}{\sin^2{\theta}}\) Hyperbola: \(r^2 = \frac{a^2b^2}{a^2\sin^2{\theta} - b^2\cos^2{\theta}}\)

Step-by-step solution

Write the general equation of an ellipse

The general equation of an ellipse is given by: (1) \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Convert the general equation into polar coordinates

To convert the ellipse equation (1) into polar coordinates, we use the following conversion formulas: \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\) Substitute these formulas into the ellipse equation (1): \(\frac{(r\cos{\theta})^2}{a^2} + \frac{(r\sin{\theta})^2}{b^2} = 1\)

Simplify the polar equation

Now we can simplify the equation: \(r^2\left(\frac{\cos^2{\theta}}{a^2} + \frac{\sin^2{\theta}}{b^2}\right) = 1\) Thus, the polar equation of the ellipse is: \(r^2 = \frac{a^2b^2}{a^2 \sin^2{\theta} + b^2 \cos^2{\theta}}\) #Parabola#

Write the general equation of a parabola

The general equation of a parabola is given by: (2) \(y^2 = 4ax\)

Convert the general equation into polar coordinates

To convert the parabola equation (2) into polar coordinates, we use the following conversion formulas: \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\) Substitute these formulas into the parabola equation (2): \((r\sin{\theta})^2 = 4a(r\cos{\theta})\)

Simplify the polar equation

Now we can simplify the equation: \(r^2\sin^2{\theta} = 4ar\cos{\theta}\) Thus, the polar equation of the parabola is: \(r = \frac{4a\cos{\theta}}{\sin^2{\theta}}\) #Hyperbola#

Write the general equation of a hyperbola

The general equation of a hyperbola is given by: (3) \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

Convert the general equation into polar coordinates

To convert the hyperbola equation (3) into polar coordinates, we use the following conversion formulas: \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\) Substitute these formulas into the hyperbola equation (3): \(\frac{(r\cos{\theta})^2}{a^2} - \frac{(r\sin{\theta})^2}{b^2} = 1\)

Simplify the polar equation

Now we can simplify the equation: \(r^2\left(\frac{\cos^2{\theta}}{a^2} - \frac{\sin^2{\theta}}{b^2}\right) = 1\) Thus, the polar equation of the hyperbola is: \(r^2 = \frac{a^2b^2}{a^2\sin^2{\theta} - b^2\cos^2{\theta}}\) Now, we have the polar equations for all three conic sections. Ellipse: \(r^2 = \frac{a^2b^2}{a^2 \sin^2{\theta} + b^2 \cos^2{\theta}}\) Parabola: \(r = \frac{4a\cos{\theta}}{\sin^2{\theta}}\) Hyperbola: \(r^2 = \frac{a^2b^2}{a^2\sin^2{\theta} - b^2\cos^2{\theta}}\)