Discriminant of a Conic Section
The general equation of a conic section is a second-degree equation in two independent variables (say ) which can be written as
There are several ways of classifying conic sections using the above general equation with the help of the discriminant of this equation:
which describes the nature of the conic section. If is zero, it represents a degenerate conic section; otherwise, it represents a non-degenerate conic section. This wiki page will give detailed information about the discriminant of a conic section.
Nature of Conic Section
Any second-degree curve equation can be written as
or
where
Now, let
then the type of conic section that the above equation represents can be found using the discriminant of the equation, which is given by for or equivalently, for
The various conditions regarding the quadratic discriminant are as follows:
If
If the equation represents two distinct real lines.
If the equation represents parallel lines.
If the equation represents non-real lines.
If
If it represents a hyperbola and a rectangular hyperbola
If the equation represents a parabola.
If the equation represents a circle or an ellipse For a real ellipse,
What type of conic section does the following equation represent:
Equating the given equation with at the top of the page, we have The discriminant for this equation is
Rewriting the given equation in the form of we have Equating this with gives Now, checking we have This confirms
Therefore, since from the equation represents a parabola.
The equation represents a circle. What is
Equating the given equation with at the top of the page, we have Thus, for the equation to represent a circle, the coefficients must simultaneously satisfy the discriminant condition and also The latter condition implies Substituting this into gives which is true.
Rewriting the given equation in the form of we have Equating this with gives Now, checking we have This confirms
Therefore,
Note: The equation of the circle is
If and are nonzero real numbers, what is the condition on and such that the equation represents an ellipse that is not a circle?
Equating the given equation with at the top of the page, we have Thus, for the equation to represent an ellipse that is not a circle, the coefficients must simultaneously satisfy the discriminant condition and also The former condition is met because The latter condition is satisfied if
Rewriting the given equation in the form of we have Equating this with gives Now, checking we have This confirms
Therefore, the condition for the equation to represent an ellipse that is not a circle is
Problem Solving
Check out the following examples based on the discriminant of a conic section.
What type of conic section does the following equation represent:
Equating the given equation with at the top of the page, we have The discriminant for this equation is
Rewriting the given equation in the form of we have Equating this with gives Now, checking we have This confirms
Therefore, since from the equation represents a hyperbola.
Suppose that the equation represents a circle with radius What is
Equating the given equation with at the top of the page, we have Thus, for the equation to represent a circle, the coefficients must simultaneously satisfy the discriminant condition and also and This last condition is met if The second condition is already met because we are given Then the the discriminant condition gives which is true.
Thus, rewriting the given equation, we have where is the radius of the circle. Hence, implying
Try the following problems.