Write this in Standard form: (x-3/2)² + (y+5/3)²= 49/4

To write the given equation in standard form, we start by identifying the given equation:

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = \frac{49}{4}
\]

### Step 1: Identify the Standard Form of a Circle

The standard form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius.

### Step 2: Identify \(h\), \(k\), and \(r^2\)

Comparing with the standard form:
- \(h = \frac{3}{2}\)
- \(k = -\frac{5}{3}\)
- \(r^2 = \frac{49}{4}\)

### Step 3: Find the Radius

To find the radius \(r\), take the square root of \(r^2\):

\[
r = \sqrt{\frac{49}{4}} = \frac{\sqrt{49}}{\sqrt{4}} = \frac{7}{2}
\]

### Step 4: Write the Standard Form

The equation is already in standard form:

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = \frac{49}{4}
\]

This represents a circle with:
- Center: \(\left( \frac{3}{2}, -\frac{5}{3} \right)\)
- Radius: \(\frac{7}{2}\)

### Final Answer

The equation in standard form is:

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = \frac{49}{4}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you convert an equation to standard form?
2. What are the steps to find the center and radius from a circle equation?
3. How can you write an equation in general form given its standard form?
4. What is the meaning of each parameter in the standard form of a circle?
5. How do you graph a circle given its standard form equation?

**Tip:** Remember that the standard form for a circle always involves the square of the radius on the right side.

Learn how to convert the circle equation (x-3/2)^2 + (y+5/3)^2 = 49/4 to its standard form. This problem involves identifying the center and radius of a circle from its given equation. Suitable for Grades 10-12.

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