Let's solve each of the three problems step by step and then write the equations of the ellipses in standard form.

1. Center at (-3,-7), Vertical Major Axis of Length 20, Minor Axis of Length 12

Given:

• Center: $(h, k) = (-3, -7)$
• Length of Major Axis: $2a = 20$, so $a = 10$
• Length of Minor Axis: $2b = 12$, so $b = 6$
• Since the major axis is vertical, the standard form of the ellipse is:

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

Substituting the values:

$\frac{(x + 3)^2}{6^2} + \frac{(y + 7)^2}{10^2} = 1$

Simplifying:

$\frac{(x + 3)^2}{36} + \frac{(y + 7)^2}{100} = 1$

2. A Focus at (-9,15) and a Covertex at (1,10) with Vertical Major Axis

Given:

• Focus: $(h, k+c) = (-9, 15)$
• Covertex: $(h+b, k) = (1, 10)$
• Vertical Major Axis implies the equation form is:

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

Let's identify $h$, $k$, $a$, and $b$:

• The center, $(h, k)$, lies midway between the focus and the corresponding point on the axis: $k + c = 15 \text{ and } k = 10 \text{ so } c = 5.$

Thus, $h = -9$ and $k = 10$.

• The distance between the covertex and the center gives us $b$: $h + b = 1 \text{ and } h = -9 \text{ so } b = 10.$

• Now to find $a$, recall the relationship: $a^2 = b^2 + c^2.$

Since $c = 5$ and $b = 10$, we calculate $a$: $a^2 = 10^2 + 5^2 = 100 + 25 = 125,$ $a = \sqrt{125}.$

Therefore, the equation is: $\frac{(x+9)^2}{100} + \frac{(y-10)^2}{125} = 1$

3. A Vertex at (-3,-18) and a Covertex at (-12,-7) Major Axis is Either Horizontal or Vertical

Given:

• Vertex: $(h, k-a) = (-3, -18)$

• Covertex: $(h-b, k) = (-12, -7)$

• Possible forms of the equation:

• If the major axis is vertical: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

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

Let's identify $h$, $k$, $a$, and $b$:

• The center $(h, k)$ lies between the vertex and covertex.

• We determine the center $(h, k)$ by averaging the coordinates. $h = -3, k = -7$

• The distance between the vertex and the center gives us $a$. $a = |-18 + 7| = 11$

• The distance between the covertex and the center gives us $b$: $b = |-12 + 3| = 9.$

If the major axis is vertical: $\frac{(x+3)^2}{9^2} + \frac{(y+7)^2}{11^2} = 1$ $\frac{(x+3)^2}{81} + \frac{(y+7)^2}{121} = 1$

Alternatively, if the major axis is horizontal: $\frac{(x+3)^2}{11^2} + \frac{(y+7)^2}{9^2} = 1$ $\frac{(x+3)^2}{121} + \frac{(y+7)^2}{81} = 1$

To determine the orientation:

• Since the major axis is either horizontal or vertical, you can compare the positions of the vertex and covertex to determine the correct axis.

Would you like the graph for these ellipses or more detailed explanations?

Related Questions:

1. How to derive the standard form of an ellipse given its vertices and foci?
2. What is the relationship between the semi-major axis, semi-minor axis, and the distance between foci in an ellipse?
3. How do you convert the equation of an ellipse from general form to standard form?
4. What is the significance of the orientation of the major and minor axes in an ellipse?
5. How can we determine if an ellipse is stretched more vertically or horizontally?

Tip: In ellipses, the sum of the distances from any point on the ellipse to the two foci is constant. This is a key property used to derive its equation.