Here's an undergraduate-level problem on conic sections, specifically focusing on the ellipse. This will involve identifying the equation of an ellipse from general information and solving it step by step.

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Problem:

Given the foci of an ellipse at points $F_1(2, 3)$ and $F_2(8, 3)$, and the length of the major axis is 10 units, find the equation of the ellipse.

Step-by-Step Solution:

Step 1: Identify the basic properties of an ellipse.

For an ellipse, the general equation is:

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

Where:

• $(h, k)$ is the center of the ellipse.
• $a$ is the semi-major axis (half of the length of the major axis).
• $b$ is the semi-minor axis (half of the length of the minor axis).
• The foci are located along the major axis, and the relationship between $a$, $b$, and the distance from the center to each focus $c$ is given by:

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

Step 2: Find the center of the ellipse.

The foci are given as $F_1(2, 3)$ and $F_2(8, 3)$.

• The center of the ellipse is the midpoint between the two foci. The midpoint formula is:

$\text{Center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substitute the coordinates of $F_1(2, 3)$ and $F_2(8, 3)$:

$\text{Center} = \left( \frac{2 + 8}{2}, \frac{3 + 3}{2} \right) = (5, 3)$

Thus, the center of the ellipse is at $(5, 3)$.

Step 3: Determine the value of $a$.

The problem states that the length of the major axis is 10 units.

• The semi-major axis $a$ is half of the length of the major axis, so:

$a = \frac{10}{2} = 5$

Step 4: Find the distance between the foci.

The foci $F_1(2, 3)$ and $F_2(8, 3)$ lie on the horizontal axis (the x-axis). The distance between the foci is:

$\text{Distance between the foci} = |8 - 2| = 6$

Step 5: Calculate the value of $c$.

The distance between the foci is related to the semi-major axis $a$ and the semi-minor axis $b$ through the relationship:

$2c = 6 \quad \Rightarrow \quad c = \frac{6}{2} = 3$

Now, we can use the equation $c^2 = a^2 - b^2$ to find $b$. Substituting $c = 3$ and $a = 5$:

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

Solving for $b^2$:

$b^2 = 25 - 9 = 16 \quad \Rightarrow \quad b = 4$

Step 6: Write the equation of the ellipse.

Now that we know:

• Center $(h, k) = (5, 3)$
• Semi-major axis $a = 5$
• Semi-minor axis $b = 4$

The standard form of the ellipse equation is:

$\frac{(x - 5)^2}{5^2} + \frac{(y - 3)^2}{4^2} = 1$

Simplifying:

$\frac{(x - 5)^2}{25} + \frac{(y - 3)^2}{16} = 1$

Final Answer:

The equation of the ellipse is:

$\frac{(x - 5)^2}{25} + \frac{(y - 3)^2}{16} = 1$

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Follow-Up Questions:

1. How would the equation change if the foci were vertically aligned instead of horizontally?
2. What would happen to the shape of the ellipse if the distance between the foci increased but the length of the major axis remained the same?
3. If we had a hyperbola with the same foci, how would its equation differ from that of the ellipse?
4. How do you determine the length of the minor axis $b$ if only the eccentricity $e$ of the ellipse is known?
5. What happens to the equation of the ellipse if the center is at the origin?

Tip:

When working with ellipses, remember that the relationship $c^2 = a^2 - b^2$ is key to connecting the semi-major axis, semi-minor axis, and the distance from the center to the foci.