25-28 Polar Equation for a Hyperbola A polar equation of a conic is given.(a) Show that the conic is a hyperbola, and sketch its graph.(b) Find the vertices and directrix, and indicate them on the graph.(c) Find the center of the hyperbola, and sketch the asymptotes.
The polar equation is a hyperbola and draw the graph of the equation.
The general equation of conic in polar form with one focus at origin and eccentricity is given by as below.
The conditions for different conic section for above formula follows as below.
a) It is a parabola if .
b) It is an ellipse if .
c) It is hyperbola if .
Consider the given equation,
Compare above equation with general equation of a conic for eccentricity .
Since, eccentricity is greater than 1, the given conic is a hyperbola.
To sketch the graph, plot the points for . Obtain the values of for these values of .
The following table is obtained.