Problem 30 - Algebra and Trigonometry

Question

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ x^{2}-2 x y+3 y^{2}=8 $$

Step-by-Step Solution

Answer

The conic is an ellipse, as confirmed by the discriminant and graphing.

Step 1: Define the Discriminant

The general equation of a conic is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here, \( A = 1 \), \( B = -2 \), and \( C = 3 \). The discriminant \( \Delta \) is given by \( B^2 - 4AC \).

Step 2: Calculate the Discriminant

Substitute \( A = 1 \), \( B = -2 \), and \( C = 3 \) into the discriminant formula: \[ \Delta = (-2)^2 - 4(1)(3) = 4 - 12 = -8 \]

Step 3: Identify the Conic

For conics, the discriminant determines the type: - If \( \Delta > 0 \), the conic is a hyperbola.- If \( \Delta = 0 \), the conic is a parabola.- If \( \Delta < 0 \), the conic is an ellipse. Since \( \Delta = -8 \), which is less than 0, the conic is an ellipse.

Step 4: Confirm by Graphing

To confirm the identification, use a graphing device or software to plot the equation \( x^{2} - 2xy + 3y^{2} = 8 \). Upon graphing, the conic should visually appear as an ellipse, confirming the result from the discriminant analysis.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a double cone (two identical right circular cones placed tip to tip) with a plane. They are incredibly important in mathematics as they form the basis of many geometrical concepts.

These sections are categorized into four main types based on the angle and location of the intersection of the cone and the plane.

• Circle: A special case of an ellipse where the plane is perpendicular to the axis of the cone.
• Ellipse: Occurs when the cutting plane intersects the cone at an angle, but not steep enough to create a parabola.
• Parabola: Formed when the plane is parallel to a generator (slant height) of the cone.
• Hyperbola: Produced when the plane intersects both naps (top and bottom) of the double cone.

Each type of conic section has unique properties and equations associated with it, used heavily in the fields of physics, astronomy, and engineering.

Ellipse

An ellipse is a type of conic section that appears as a stretched circle. It's defined as the set of points where the sum of the distances from two fixed points (foci) is constant.

The standard form of an ellipse in the Cartesian coordinate system is given by:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where

• \( (h, k) \) is the center of the ellipse.
• \( a \) and \( b \) are the distances from the center to the vertices.

The discriminant, \( \Delta = B^2 - 4AC \), helps identify the type of conic. For ellipses, the discriminant is negative (\( \Delta < 0 \)). In the case of our equation, \( \Delta = -8 \), confirming the presence of an ellipse.

Ellipses are widely encountered in real-life applications, such as planetary orbits and optics, highlighting their versatility across different scientific and mathematical domains.

Graphing Conic Sections

Graphing conic sections allows us to visualize the equations describing them, providing deeper insight into their properties and behavior.

To graph conic sections, you can use graphing software or a graphing calculator, which helps in easily plotting and confirming the type of the conic as deduced from the discriminant.

When graphing the equation \( x^2 - 2xy + 3y^2 = 8 \):

• First identify the type using the discriminant (as we've done, confirming it's an ellipse).
• Use the necessary graphing tools to import or enter the equation.
• Examine the graph to confirm visually the characteristics expected from an ellipse.

Visualizing conic sections is crucial for developing intuition about their properties and for practical applications where visual information is invaluable, like designing lenses or satellite dish antennas.