Write this in Standard form: (x-3/2)² + (y+5/3)²= 49/7

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

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = \frac{49}{7}
\]

### Step 1: Simplify the Right Side

Simplify the fraction on the right side:

\[
\frac{49}{7} = 7
\]

Now the equation becomes:

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = 7
\]

### Step 2: 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 3: Identify \(h\), \(k\), and \(r^2\)

Comparing with the standard form:
- \(h = \frac{3}{2}\)
- \(k = -\frac{5}{3}\)
- \(r^2 = 7\), so \(r = \sqrt{7}\)

Thus, the standard form of the equation is already:

\[
\left( x - \frac{3}{2} \right)^2 + \left( y + \frac{5}{3} \right)^2 = 7
\]

This is the equation of a circle with center \(\left( \frac{3}{2}, -\frac{5}{3} \right)\) and radius \(\sqrt{7}\).

Would you like more details or have any questions?

### Related Questions:
1. What is the general form of a circle's equation?
2. How do you find the radius of a circle given its equation?
3. How do you convert a circle's equation from standard to general form?
4. What is the significance of the center and radius in a circle's equation?
5. How do you find the distance between two points?

**Tip:** Remember that the standard form for a circle's equation is always \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Learn how to convert the circle equation (x - 3/2)^2 + (y + 5/3)^2 = 49/7 to its standard form. This problem covers key concepts including circle equations, standard forms, and identifying circle properties. Suitable for students in Grades 9-12.

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