Question

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic.

Answer 1

Step 1:

(a)

The polar equation of the conic is .

Convert the given polar equation into standard form of the polar equation .

Take out 6 as common from the denominator.

Now compare the above equation with standard form.

and .

The eccentricity of the conic equation is .

Answer 2

Contd...

Step 2:

(b)

The eccentricity of the conic .

As eccentricity , the given conic section is a ellipse.

Step 3:

(c)

 The ellipse equation is .

 

The directrix is parallel to the polar axis .

The value in the numerator is .

Substitute  in the .

So the directrix of the ellipse is .

Answer 3

Contd...

Step 4:

(d)

The polar equation is .

Construct a table for different values of .

 

Answer 4

Contd...

Graph:

(1) Graph the polar co-ordinates.

(2) Plot the points.

(3) Connect the points to a smooth curve.

Solution :

(a) The eccentricity of the conic equation is .

(b) The given conic section is a ellipse.

(c) The directrix of the ellipse is .

(d)