https://file.aiquickdraw.com/mathbot/file/1725546643102tdenfonr.jpg

The image contains a set of four mathematical expressions, all representing conic sections, with the instructions asking to "Convert the following standard equations to general equations."

Here are the four equations visible in the image:

1. \(\frac{y^2}{16} + \frac{x^2}{20} = 1\)
2. \(\frac{(x-5)^2}{9} + \frac{(y+3)^2}{16} = 1\)
3. \(\frac{x^2}{49} - \frac{(y+6)^2}{25} = 1\)
4. \(\frac{(x+2)^2}{4} - \frac{(y-1)^2}{4} = 1\)

Let me walk you through the process of converting each of these conic equations from standard form to general form:

### 1. \(\frac{y^2}{16} + \frac{x^2}{20} = 1\)
This is the equation of an ellipse. Multiply through by 16 and 20 (the denominators) to eliminate the fractions:
\[
20y^2 + 16x^2 = 320
\]
This is the general form of the equation for this ellipse.

### 2. \(\frac{(x-5)^2}{9} + \frac{(y+3)^2}{16} = 1\)
This is also an ellipse. Multiply both sides by 144 (LCM of 9 and 16) to eliminate the fractions:
\[
16(x - 5)^2 + 9(y + 3)^2 = 144
\]
Now expand both terms:
\[
16(x^2 - 10x + 25) + 9(y^2 + 6y + 9) = 144
\]
\[
16x^2 - 160x + 400 + 9y^2 + 54y + 81 = 144
\]
Now combine like terms:
\[
16x^2 + 9y^2 - 160x + 54y + 481 = 144
\]
Thus, the general form is:
\[
16x^2 + 9y^2 - 160x + 54y + 337 = 0
\]

### 3. \(\frac{x^2}{49} - \frac{(y+6)^2}{25} = 1\)
This is the equation of a hyperbola. Multiply both sides by 1225 (LCM of 49 and 25) to eliminate the fractions:
\[
25x^2 - 49(y + 6)^2 = 1225
\]
Now expand the second term:
\[
25x^2 - 49(y^2 + 12y + 36) = 1225
\]
\[
25x^2 - 49y^2 - 588y - 1764 = 1225
\]
Now combine like terms:
\[
25x^2 - 49y^2 - 588y - 2989 = 0
\]
Thus, the general form is:
\[
25x^2 - 49y^2 - 588y - 2989 = 0
\]

### 4. \(\frac{(x+2)^2}{4} - \frac{(y-1)^2}{4} = 1\)
This is also a hyperbola. Multiply both sides by 4 to eliminate the fractions:
\[
(x + 2)^2 - (y - 1)^2 = 4
\]
Now expand both terms:
\[
(x^2 + 4x + 4) - (y^2 - 2y + 1) = 4
\]
\[
x^2 + 4x + 4 - y^2 + 2y - 1 = 4
\]
Now combine like terms:
\[
x^2 - y^2 + 4x + 2y + 3 = 4
\]
Thus, the general form is:
\[
x^2 - y^2 + 4x + 2y - 1 = 0
\]

Would you like further details on any step or help with another part? Here are five related questions to deepen your understanding:

1. How can you recognize the conic section (ellipse, hyperbola, etc.) from the standard form of an equation?
2. What are the steps to convert a general conic equation back into standard form?
3. How can you graph an ellipse or a hyperbola from its general equation?
4. What transformations occur when terms in the standard form are modified (e.g., shifting the center)?
5. How can the discriminant of a conic equation help determine the type of conic section?

**Tip:** For hyperbolas, the sign between the \(x^2\)

Convert the following standard conic section equations to general equations:
1) \(\frac{y^2}{16} + \frac{x^2}{20} = 1\)
2) \(\frac{(x-5)^2}{9} + \frac{(y+3)^2}{16} = 1\)
3) \(\frac{x^2}{49} - \frac{(y+6)^2}{25} = 1\)
4) \(\frac{(x+2)^2}{4} - \frac{(y-1)^2}{4} = 1\)

Learn how to convert standard conic section equations to their general form. This problem covers ellipses and hyperbolas and requires knowledge of expanding binomials and combining like terms. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation