Question

For the following equations, determine which of the conic sections is described. $$ 34 x^{2}-24 x y+41 y^{2}-25=0 $$

Short answer

The given equation describes an ellipse.

Step-by-step solution

Recall Conic Sections

A conic section can be one of the following: circle, ellipse, parabola, or hyperbola. The general equation for a conic section is given by: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The presence of the \(Bxy\) term indicates a rotation of the conic section.

Identify Coefficients

Identify the coefficients from the given equation \[ 34x^{2} - 24xy + 41y^{2} - 25 = 0 \]:- \(A = 34\)- \(B = -24\)- \(C = 41\)- \(D = 0\) (since no \(x\) term exists)- \(E = 0\) (since no \(y\) term exists)- \(F = -25\)

Calculate the Discriminant

The discriminant \(\Delta\) is used to determine the type of conic section. It is calculated as:\[ \Delta = B^2 - 4AC \]Substitute the identified coefficients:\[ \Delta = (-24)^2 - 4(34)(41) \]

Solve the Discriminant

Calculate \( \Delta \):\[ (-24)^2 = 576 \]\[ 4 \times 34 \times 41 = 5576 \]\[ \Delta = 576 - 5576 = -5000 \]

Determine the Conic Section

Using the value of \( \Delta \):- If \( \Delta = 0 \), it is a parabola.- If \( \Delta > 0 \), it is a hyperbola.- If \( \Delta < 0 \), it is an ellipse.Since \( \Delta = -5000 \lt 0 \), the conic section is an ellipse.

Key concepts

Ellipse

An ellipse is a type of conic section. It resembles an elongated circle or an oval shape. The ellipse is one of the four possible shapes formed when a plane cuts through a cone in different ways. An important property of an ellipse is that it has two axes: the major axis (longest diameter) and the minor axis (shortest diameter). These axes intersect at the center of the ellipse.

Some core characteristics of an ellipse include:

• The sum of the distances from any point on the ellipse to two fixed points, called foci, is constant.
• An ellipse is symmetric around its major and minor axes.

Ellipses appear in various fields ranging from astronomy (planetary orbits) to engineering (paths of certain mechanical parts). Understanding their mathematical foundations enables us to model and analyze these phenomena effectively.

Discriminant in Conic Sections

The discriminant is a helpful tool in determining the type of conic section an equation represents. The general form of a conic section equation includes terms for both the square of the variables and a product of these variables, in the form: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]The key part of this equation in determining the type is the discriminant, given by:\[\Delta = B^2 - 4AC\]

Here's how the discriminant helps identify the conic section:

• If \(\Delta = 0\), the conic section is a parabola.
• If \(\Delta > 0\), the conic section is a hyperbola.
• If \(\Delta < 0\), the conic section is an ellipse. This was the case in our equation where \(\Delta = -5000\).

Using the discriminant is a straightforward and reliable method to quickly classify conic sections.

Conic Section Identification

Conic sections are a family of curves derived from intersecting a plane with a double-napped cone. Depending on the angle at which the plane intersects the cone, different shapes are produced: a circle, an ellipse, a parabola, or a hyperbola. Correctly identifying conic sections is important in geometry and calculus as it helps in understanding the properties and equations associated with each shape.

Here's a quick guide to identifying conic sections using their equations:

• If the equation lacks the \(xy\) term and the coefficients of \(x^2\) and \(y^2\) are equal, it's a circle.
• If the equation includes an \(xy\) term, it's oriented such that an ellipse, parabola, or hyperbola may be present, in which case the discriminant, \(\Delta = B^2 - 4AC\), determines its type.
• Further classify by:
  • Circle: no \(xy\) term and equal coefficients.
  • Ellipse: \(\Delta < 0\)
  • Parabola: \(\Delta = 0\)
  • Hyperbola: \(\Delta > 0\)

The equation in our example resulted in an ellipse, confirming its nature by the discriminant calculation.