6.5 Q-1

Question Statement

Find the equations of ellipses for the given data and sketch their graphs.

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Background and Explanation

An ellipse is defined as the locus of all points such that the sum of their distances to two fixed points (foci) is constant. The standard form of the ellipse equation is:

1. Horizontal Major Axis:
   $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, , a > b$

2. Vertical Major Axis:
   $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, , a > b$

Here:

• a is the semi-major axis,
• b is the semi-minor axis,
• c is the distance from the center to each focus,
• Relationship: $c^2 = a^2 - b^2$.

The eccentricity $e$ is calculated as $e = \frac{c}{a}$, where $e < 1$.

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Solution

1: Foci $( \pm 3, 0)$, Minor Axis Length 10

Given:

• Foci at $(\pm 3, 0)$,
• Minor axis length = 10, so $b = 5$,
• $c = 3$.

We calculate $a^2$ using $c^2 = a^2 - b^2$:

$$c^2 = a^2 - b^2 \quad \Rightarrow \quad 3^2 = a^2 - 5^2 \quad \Rightarrow \quad 9 = a^2 - 25 \quad \Rightarrow \quad a^2 = 34.$$

The equation of the ellipse is:

$$\frac{x^2}{34} + \frac{y^2}{25} = 1$$

Sketch:
Placeholder for the diagram

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2: Foci $(0, -1)$ and $(0, -5)$, Major Axis Length 6

Given:

• Foci at $(0, -1)$ and $(0, -5)$,
• Major axis length = 6, so $a = 3$,
• The center is $(0, -3)$, and $c = 2$.

We calculate $b^2$ using $c^2 = a^2 - b^2$:

$$c^2 = a^2 - b^2 \quad \Rightarrow \quad 2^2 = 3^2 - b^2 \quad \Rightarrow \quad 4 = 9 - b^2 \quad \Rightarrow \quad b^2 = 5.$$

Since the major axis is vertical, the equation of the ellipse is:

$$\frac{x^2}{5} + \frac{(y + 3)^2}{9} = 1$$

Sketch:
Placeholder for the diagram

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3: Foci $(-3\sqrt{3}, 0)$, Vertices $( \pm 6, 0)$

Given:

• Foci at $(-3\sqrt{3}, 0)$,
• Vertices at $( \pm 6, 0)$, so $a = 6$.

Calculate $c$ and $b^2$:

$$c = 3\sqrt{3}, \quad c^2 = (3\sqrt{3})^2 = 27,$$ $$c^2 = a^2 - b^2 \quad \Rightarrow \quad 27 = 36 - b^2 \quad \Rightarrow \quad b^2 = 9.$$

The equation of the ellipse is:

$$\frac{x^2}{36} + \frac{y^2}{9} = 1$$

Sketch:
Placeholder for the diagram

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Key Formulas or Methods Used

1. Relationship:
   $c^2 = a^2 - b^2$

2. Horizontal Ellipse Equation:
   $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

3. Vertical Ellipse Equation:
   $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

4. Eccentricity:
   $e = \frac{c}{a}$

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Summary of Steps

1. Identify the type of ellipse (horizontal or vertical) based on the given data.
2. Use $c^2 = a^2 - b^2$ to find the missing parameter.
3. Substitute the values into the standard equation for the ellipse.
4. Sketch the graph, marking key features like the center, foci, and vertices.

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