Question

The set of all values of a for which equation \(\frac{x^{2}}{8-a}+\frac{y^{2}}{a-2}=1\) will represent an ellipse, is (1) \((1,4)\) (2) \((-\infty, 2) \cup(8, \infty)\) (3) \((2,8)-\\{5\\}\) (4) \((2,8)\)

Short answer

Option (4): \((2, 8)\)

Step-by-step solution

Identify the general form of an ellipse

Recall that the standard form of an ellipse centered at the origin is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where both \(a^2\) and \(b^2\) must be positive.

Compare given equation to general form of an ellipse

The given equation is \[ \frac{x^2}{8-a} + \frac{y^2}{a-2} = 1 \]. Compare this to \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. For this to represent an ellipse, both \(8 - a\) and \(a - 2\) must be positive.

Set conditions for the coefficients to be positive

Set the inequalities: \[ 8 - a > 0 \ a - 2 > 0 \]. Solving these gives: \[ a < 8 \ a > 2 \]. Therefore, the combined range for \(a\) is \[ 2 < a < 8 \].

Identify any exclusions within the valid range

Check if there are any values within \(2 < a < 8\) that should be excluded. If either \(8 - a = 0\) or \(a - 2 = 0\), the equation would be invalid. However, since we have already derived that \(2 < a < 8\), there are no exclusions within this interval.

Verify the correct set from given options

From the derived range, the interval is \(2 < a < 8\). Compare this to the provided options and identify the matching set. Option (4) matches correctly: \((2, 8)\).

Key concepts

Conic Sections

Conic sections are the shapes you get when you slice through a cone. They include circles, ellipses, parabolas, and hyperbolas. Each shape has a specific equation form and properties.

For ellipses, the general equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where both \(a^2\) and \(b^2\) must be positive. This reflects that both the major and minor axes are defined, ensuring the shape is closed and oval.

In the case of the given problem, the equation \( \frac{x^2}{8-a} + \frac{y^2}{a-2} = 1 \) must be manipulated to determine the conditions under which it defines an ellipse. By establishing the inequalities where the denominators are positive, we ensure the equation adheres to the general ellipse definition.

Understanding conic sections is crucial as they model various real-world phenomena from planetary orbits (ellipses), paths of projectiles (parabolas), to the layout of certain optical systems (hyperbolas).

Inequalities in Mathematics

Inequalities are mathematical statements that involve expressions not being entirely equal. They use signs like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

Solving inequalities often involves finding the range of values for a variable that satisfies the inequality. For example, in the given problem, we derived two inequalities for 'a':
\[ 8 - a > 0 \] means that \( a < 8 \).
\[ a - 2 > 0 \] means that \( a > 2 \).
Combining these inequalities, we get the range \[ 2 < a < 8 \].

Making sure these conditions are met ensures that the equation represents an ellipse. Working with inequalities helps in bounding variables within certain regions, aiding in areas like optimization and solving constraints within various mathematical problems.

Graph Theory

Graph theory isn't directly mentioned in the problem, but understanding how to analyze and sketch the graphs of equations is critical.

In the context of the ellipse equation, graphing involves plotting the relationship described by \[ \frac{x^2}{8-a} + \frac{y^2}{a-2} = 1 \]. Once we have valid intervals for 'a', we can plot different ellipses by differing 'a' values within the range.

Techniques from graph theory help explain how different shapes emerge from equations. They offer insights into symmetries, focal points, and can even predict how changes in equations alter their graphical representations.
Applying graph theory in more advanced ways can interlink nodes and edges, revealing deeper structures in mathematical modeling and solutions in networked systems.