Question

For the following equations, determine which of the conic sections is described. $$ x^{2}+4 x y-2 y^{2}-6=0 $$

Short answer

The equation describes a hyperbola.

Step-by-step solution

Identify the Type

Conic sections are typically equations of the form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]By the coefficients, you can determine the type: - If \( B^2 - 4AC < 0 \), it's an ellipse.- If \( B^2 - 4AC = 0 \), it's a parabola.- If \( B^2 - 4AC > 0 \), it's a hyperbola.

Match the Equation to the General Form

The given equation is:\[ x^2 + 4xy - 2y^2 - 6 = 0 \]This can align to the general form where:\( A = 1 \), \( B = 4 \), \( C = -2 \).

Calculate the Discriminant

Now calculate \( B^2 - 4AC \):\[ B^2 - 4AC = 4^2 - 4(1)(-2) \]\[ = 16 + 8 \]\[ = 24 \].

Determine the Conic Section

Since \( B^2 - 4AC = 24 > 0 \), according to the conditions, the conic section is a hyperbola.

Key concepts

Ellipse

An ellipse is a set of all points in a plane where the sum of the distances from two fixed points, called foci, is constant. You can think of it as a stretched circle, appearing oval in shape. The general equation for an ellipse can often be written as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), where the condition \( B^2 - 4AC < 0 \) holds true.

A simple way to visualize an ellipse is to imagine a regular circle that's been stretched along one axis. There are a few key properties to remember:

• The longer "diameter" of the ellipse is called the major axis, while the shorter one is the minor axis.
• The center is the midpoint of these axes.
• When \( A = C \) and \( B = 0\), the ellipse becomes a circle—an ellipse with equal axes.

Ellipses have certain applications in real life, like planetary orbits where planets move around the sun in elliptical paths. This unique property makes ellipses very important in astronomy and physics. Understanding these principles will help you identify ellipses when dealing with conic sections.

Parabola

A parabola is a symmetric curve formed by all points at an equal distance from a fixed point called the focus and a fixed line called the directrix. Parabolas open in different directions (up, down, left, or right) depending on the equation form. The general form for a conic section is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). For parabolas, the discriminant condition \( B^2 - 4AC = 0 \) applies.

A common representation of a parabola is \( y = ax^2 + bx + c \), which opens upwards or downwards, depending on the sign of \( a \). Here are some interesting details about parabolas:

• The vertex of the parabola is the point where it changes direction, and it's either a maximum or a minimum point of the function.
• The axis of symmetry is a line that passes through the vertex, dividing the parabola into two mirror-image halves.
• In real life, parabolas are often seen in satellite dishes, headlights, and even the paths of projectile motion.

These qualities make parabolas a fascinating component of conic sections, especially given their symmetrical properties and diverse real-world applications.

Hyperbola

A hyperbola consists of two separate curves called branches, and it represents all points where the difference of the distances to two foci is constant. Identifiable by its unmistaken double curve, a hyperbola appears when the equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) holds the condition \( B^2 - 4AC > 0 \).

Hyperbola can be complex to visualize but are crucially important. Let's break down its specific features:

• The transverse axis is the line segment that connects the two vertices, which are the closest points on each branch.
• The center of the hyperbola is the midpoint of the transverse axis.
• Hyperbolas have asymptotes—lines that the branches get infinitely close to, but never actually meet.

Like ellipses and parabolas, hyperbolas also make significant appearances in the real world. They are used in navigation systems, such as in GPS devices. This makes understanding hyperbolas critical, not only in mathematics but in technology and physics too.