Problem 17 - Advanced Engineering Mathematics

Question

Identifying \(A=B=C=1,\) we compute \(B^{2}-4 A C=-3 < 0 .\) The equation is elliptic.

Step-by-Step Solution

Answer

The equation represents an ellipse because \( B^2 - 4AC < 0 \).

Step 1: Identify Given Values

We are given that \( A = B = C = 1 \). This means we have the coefficients for identifying the type of conic section.

Step 2: Compute \( B^2 - 4AC \)

The formula \( B^2 - 4AC \) helps determine the type of conic. For our values, this computes to \( 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3 \).

Step 3: Determine the Significance of \( B^2 - 4AC \)

The value \( B^2 - 4AC = -3 \) is less than 0, which indicates that the conic section is an ellipse.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. Depending on the angle at which the plane intersects the cone, we can get different shapes: circles, ellipses, parabolas, or hyperbolas. These shapes are significant in various mathematical and physical applications.
To classify a conic section, mathematicians use the general quadratic equation: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The type of conic section is determined by the discriminant of the conic, which is defined by the expression \( B^2 - 4AC \). Each conic section has distinct features based on the value:

• If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), the conic is a parabola.
• If \( B^2 - 4AC < 0 \), the conic is an ellipse, as in our given exercise.

Understanding conic sections helps students to recognize patterns and solve real-world problems efficiently.

Ellipses

Ellipses are one form of conic sections characterized by the discriminant \( B^2 - 4AC < 0 \), as in our exercise. An ellipse is a set of points on a plane, such that the sum of the distances of each point from two fixed points (foci) is constant.
The standard equation of an ellipse centered at the origin \((h, k)\) takes the form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
Some important properties of ellipses include:

• The longest diameter is called the major axis.
• The shortest diameter is called the minor axis.
• The points where the ellipse intersects its axes are called the vertices.

These properties are fundamental in various fields such as astronomy, where planets have elliptical orbits, and engineering, where ellipses are used in designing reflective properties of lenses.

Algebraic Equations

Algebraic equations relate variables and constants using algebraic operations (addition, subtraction, multiplication, and division). In the context of conic sections, algebraic equations represent the curves on a graph.
The general form of an algebraic equation for conics is:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Here, each term represents an element of the ellipse, circle, parabola, or hyperbola, depending on the discriminant \( B^2 - 4AC \).
To determine the type of conic section from the equation, evaluate the coefficients \( A, B, \) and \( C \) using the discriminant

• For an ellipse, the key condition is \( B^2 - 4AC < 0 \).
• For specifically horizontal or vertical ellipses, \( B = 0 \).

Algebraic equations not only classify conic sections but also provide the framework for solving geometric problems in coordinate geometry.