Problem 60 - College Algebra

Question

For the following exercises, determine the value of \(k\) based on the given equation. Given \(6 x^{2}+12 x y+k y^{2}+16 x+10 y+4=0\) find \(k\) for the graph to be an ellipse.

Step-by-Step Solution

Answer

The value of \(k\) must be greater than 6.

Step 1: Identify the Standard Form of an Ellipse

The general conic section equation is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For the equation to represent an ellipse, it must satisfy the condition \(B^2 - 4AC < 0\).

Step 2: Compare with Standard Conic Form

Identify \(A = 6\), \(B = 12\), and \(C = k\) from the given equation \(6x^2 + 12xy + ky^2 + 16x + 10y + 4 = 0\).

Step 3: Substitute into Discriminant Condition

Substitute the values of \(A\), \(B\), and \(C\) into the condition \(B^2 - 4AC < 0\). This gives: \(12^2 - 4 \cdot 6 \cdot k < 0\).

Step 4: Simplify the Inequality

Calculate \(12^2\) which is \(144\). Therefore, the inequality becomes \(144 - 24k < 0\).

Step 5: Solve for k

Rearrange the inequality \(144 - 24k < 0\) to solve for \(k\). This gives \(144 < 24k\), which simplifies to \(k > 6\).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. They are classified into four main types: circles, ellipses, parabolas, and hyperbolas. Each type of conic section has distinct properties and equations that define its shape.

• Circles: A circle is a special case of an ellipse where the major and minor axes are equal. The equation is generally of the form \(x^2 + y^2 = r^2\).


• Ellipses: These have the general equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) where \(B^2 - 4AC < 0\).


• Parabolas: These take the form \(Ax^2 + Dx + Ey + F = 0\) and satisfy \(B^2 - 4AC = 0\).


• Hyperbolas: Their equation is similar to the ellipse but with \(B^2 - 4AC > 0\).

Understanding these types helps in identifying the shape of the graph based on its equation. When dealing with conic sections, especially ellipses, it's important to pay attention to the coefficients in the equation.

Discriminant Condition

The discriminant condition is a mathematical expression used to determine the specific type of conic section represented by a quadratic equation in two variables. This expression is particularly useful for equations of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
The key discriminant expression for conics is given by \(B^2 - 4AC\). Here is how it can be interpreted:

• Ellipse: If \(B^2 - 4AC < 0\), the conic is an ellipse, indicating a negative discriminant value.


• Parabola: If \(B^2 - 4AC = 0\), the conic represents a parabola with zero discriminant.


• Hyperbola: If \(B^2 - 4AC > 0\), the conic is a hyperbola, also known as having a positive discriminant.

This condition helps in quickly categorizing conics by just evaluating the mathematical relationship between the coefficients. It forms a crucial step in converting the general conic equation to its standard form.

Standard Form of Ellipse

The standard form of an ellipse is a simplified version of its general equation, making it easier to identify the ellipse's major and minor axes. An ellipse centered at the origin, with its axis aligned with the coordinate axes, has the standard form:

• Horizontal Major Axis: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)


• Vertical Major Axis: \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\)

In these equations:

• \((h, k)\) is the center of the ellipse.


• \(a\) is the length of the semi-major axis.


• \(b\) is the length of the semi-minor axis.


• \(a\) is greater than \(b\); if \(a = b\), the ellipse is a circle.

To convert a general equation to this form, one must complete the square for both \(x\) and \(y\) terms, making sure to rearrange and manipulate the equation to isolate the x and y terms in the fractional form. Understanding the standard form allows identifying the key parameters of the ellipse directly from the equation.