https://file.aiquickdraw.com/mathbot/file/1728612437092x89d2gzz.jpg

The given equation is:

\[
49x^2 + 196x + 100y^2 + 1400y + 196 = 0
\]

This is a quadratic equation in both \(x\) and \(y\). Let's simplify this equation step by step and see if it can be factored or rewritten in a more manageable form, possibly representing a conic section (such as a circle or ellipse).

### Step 1: Grouping and factoring
We'll start by grouping the terms for \(x\) and \(y\):

\[
49(x^2 + 4x) + 100(y^2 + 14y) + 196 = 0
\]

### Step 2: Completing the square

#### For \(x\)-terms:
We have \(x^2 + 4x\), and to complete the square, we add and subtract \(\left(\frac{4}{2}\right)^2 = 4\):

\[
x^2 + 4x = (x + 2)^2 - 4
\]

#### For \(y\)-terms:
We have \(y^2 + 14y\), and to complete the square, we add and subtract \(\left(\frac{14}{2}\right)^2 = 49\):

\[
y^2 + 14y = (y + 7)^2 - 49
\]

### Step 3: Substitution
Now we can substitute these results back into the original equation:

\[
49\left[(x + 2)^2 - 4\right] + 100\left[(y + 7)^2 - 49\right] + 196 = 0
\]

Simplifying this:

\[
49(x + 2)^2 - 196 + 100(y + 7)^2 - 4900 + 196 = 0
\]

\[
49(x + 2)^2 + 100(y + 7)^2 - 4900 = 0
\]

### Step 4: Rearranging the equation
Move the constant to the other side:

\[
49(x + 2)^2 + 100(y + 7)^2 = 4900
\]

### Step 5: Dividing by 4900
Divide through by 4900:

\[
\frac{49(x + 2)^2}{4900} + \frac{100(y + 7)^2}{4900} = 1
\]

Simplifying further:

\[
\frac{(x + 2)^2}{100} + \frac{(y + 7)^2}{49} = 1
\]

### Final Form:
This is the equation of an ellipse, centered at \((-2, -7)\), with a semi-major axis of 10 (along the \(x\)-axis) and a semi-minor axis of 7 (along the \(y\)-axis).

Would you like more details on ellipses or any clarification on this problem?

Here are five related questions to expand on this topic:
1. How do you derive the general equation of a conic section from an implicit quadratic equation?
2. What are the geometric properties of an ellipse?
3. How can you calculate the foci of an ellipse given its equation?
4. How does completing the square help in recognizing conic sections?
5. What is the significance of the semi-major and semi-minor axes in an ellipse?

**Tip:** Always check for symmetry and try completing the square when dealing with quadratic equations involving two variables. It often reveals useful geometric interpretations.

This problem involves solving the quadratic equation 49x^2 + 196x + 100y^2 + 1400y + 196 = 0 and simplifying it into the standard form of an ellipse. Learn how completing the square reveals the equation of an ellipse, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation