Conic Section Equation

by

William Plummer

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The graph of a second degree equation is a conic section and can be expressed with the following equation:

There are several ways of classifying conic sections using the above general equation.

When there is no 'xy' term (B=0), the conic section can be classified by analyzing A and C:

Condition #1

then the conic section is a circle.

Click here "Save Link As" to see the graph of the circles in motion.

Slide the parameter (n) at the bottom to change the value of A and C

Also, change the values of D, E, and F to explore other circles that can be created.

Condition #2

(Either A=0 or C=0)

then the conic section is a parabola

Click here "Save Link As" to see the graph of the parabolas in motion.

Slide the parameter (n) at the bottom to change the value of A and C

Also, change the values of D, E, and F to explore other parabolas that can be created.

Condition #3

(A and C are both positive or both negative)

then the conic section is an ellipse

Click here "Save Link As" to see the graph of the ellipses in motion.

Slide the parameter (n) at the bottom to change the value of A and C

Also, change the values of D, E, and F to explore other ellipses that can be created.

Condition #4

(A and C have opposite signs)

then the conic section is a hyperbola

Click here "Save Link As" to see the graph of the hyperbolas in motion.

Slide the parameter (n) at the bottom to change the value of A and C

Also, change the values of D, E, and F to explore other hyperbolas that can be created.

**The general equation can also be used to classify the conic section by analyzing the discriminant.**

Condition #5

then the conic section is a parabola

Condition #6

then the conic section is an ellipse

(or a circle if B = 0 and A = C)

Condition #7

then the conic section is a hyperbola

Click here "Save Link As" to see the graph of the conic section and its calculated discriminant.

Change the values of A, B, C, D, E, and F to find different values for the discriminant.

Take note of the blue line representing the discriminant. (y = discriminant value)

What is the shape of the conic section when the discriminant is positive?

What is the shape of the conic section when the discriminant is negative?

What is the shape of the conic section when the discriminant is equal to 0?

What happens when you change the values of D, E, and F ?