6.5 Q-6

Question Statement

Write the equation of an ellipse where the major axis lies along the x-axis and has a length of $4\sqrt{2}$. The distance between the foci is equal to the length of the minor axis.

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Background and Explanation

For an ellipse, the standard equation is given as:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$$

where:

• $a$ is the semi-major axis.
• $b$ is the semi-minor axis.
• $c$ is the distance from the center to each focus, given by $c = \sqrt{a^2 - b^2}$.

In this problem, the major axis lies along the x-axis, and the given relationships help us determine $a$, $b$, and $c$.

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Solution

Step 1: Use the length of the major axis

The length of the major axis is $4\sqrt{2}$, which means:

$$2a = 4\sqrt{2} \quad \Rightarrow \quad a = 2\sqrt{2}.$$

Squaring $a$, we find:

$$a^2 = (2\sqrt{2})^2 = 8.$$

Step 2: Use the relationship between the minor axis and the foci

The problem states that the distance between the foci equals the length of the minor axis:

$$2c = 2b \quad \Rightarrow \quad b = c.$$

Step 3: Relate $c$, $a$, and $b$

Using the ellipse property $c^2 = a^2 - b^2$ and substituting $b = c$, we get:

$$c^2 = a^2 - c^2 \quad \Rightarrow \quad 2c^2 = a^2.$$

Substituting $a^2 = 8$:

$$2c^2 = 8 \quad \Rightarrow \quad c^2 = 4.$$

Since $b = c$, we also find:

$$b^2 = c^2 = 4.$$

Step 4: Write the equation of the ellipse

Substitute $a^2 = 8$ and $b^2 = 4$ into the standard ellipse equation:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \Rightarrow \quad \frac{x^2}{8} + \frac{y^2}{4} = 1.$$

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Key Formulas or Methods Used

1. Standard Equation of an Ellipse:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$

2. Relation Between Semi-Major and Semi-Minor Axes:

$$c^2 = a^2 - b^2.$$

3. Geometric Properties:
   • Major axis length $= 2a$.
   • Minor axis length $= 2b$.
   • Distance between foci $= 2c$.

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Summary of Steps

1. The major axis length $2a = 4\sqrt{2}$ gives $a^2 = 8$.
2. The condition $2c = 2b$ implies $b = c$.
3. Using $c^2 = a^2 - b^2$, solve for $b^2 = 4$.
4. Substitute $a^2$ and $b^2$ into the standard ellipse equation:

$$\frac{x^2}{8} + \frac{y^2}{4} = 1.$$

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Final Equation

$$\frac{x^2}{8} + \frac{y^2}{4} = 1$$