Question

To investigate the changes in the graph of family of curves defined by the polar equations \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\) as the number changes.

Short answer

It can be concluded that shape of curve \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\) when \(0 < a < 1\) is oval, it is parabola when \(a = 1\) and it is hyperbola when \(a > 1\).

Step-by-step solution

Given data

The given polar equation is \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\).

Definition of Polar equation

The equation which defines an algebraic curve expressed in polar coordinatesis known as a polar equation.

Sketch the curve of polar equation

The given polar equation is \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\), where \(a\) is any real number.

Sketch the family of curves of polar equations \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\) for \(a = \frac{1}{2},1\) and 2 as shown in figure below.

From Figure 1, the solid lines represent \({r_1} = \frac{{1 - \cos \theta }}{{1 + \cos \theta }}\), dashed lines represent \({r_2} = \frac{{1 - \frac{1}{2}\cos \theta }}{{1 + \frac{1}{2}\cos \theta }}\) and the dotted lines represents \({r_3} = \frac{{1 - 2\cos \theta }}{{1 + 2\cos \theta }}\).

Determine the changes in the graph of family of curves

It is observed that shape of the graph of \({r_1} = \frac{{1 - \cos \theta }}{{1 + \cos \theta }}\) is oval, the graph of \({r_2} = \frac{{1 - \frac{1}{2}\cos \theta }}{{1 + \frac{1}{2}\cos \theta }}\) is parabola and graph of \({r_3} = \frac{{1 - 2\cos \theta }}{{1 + 2\cos \theta }}\) is hyperbola.

Therefore, it can be concluded that shape of curve \(r = \frac{{1 - a\cos \theta }}{{1 + a\cos \theta }}\) when \(0 < a < 1\) is oval, it is parabola when \(a = 1\) and it is hyperbola when \(a > 1\).