Conic Sections Overview Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

The general equation

2x^2 + 2y^2 + Bxy - 8 = 0

represents a circle only when

B

takes a specific value. Find that value.

1
A circle requires no $xy$ term (i.e., $B = 0$ ) and equal coefficients on $x^2$ and $y^2$ (already satisfied: both are $2$ ).
2
So $B = 0$ . The equation becomes $2x^2 + 2y^2 = 8$ , or $x^2 + y^2 = 4$ , a circle of radius $2$ .

Answer

B = 0

For a second-degree equation to represent a circle, three conditions must hold: the coefficients of

x^2

and

y^2

must be equal and have the same sign, and the coefficient of

xy

must be zero. Any nonzero

xy

term would rotate the axes, producing an ellipse or hyperbola instead.

About Conic Sections Overview

The four curves—circle, ellipse, parabola, and hyperbola—obtained by slicing a double cone with a plane at different angles.

Identify the type of conic section: [formula].

Classify the conic given by [formula].

Classify each equation: (a) [formula], (b) [formula], (c) [formula].