Find the standard form of the equation of the circle having the following​ properties:
Center at the origin
Containing the point left parenthesis 2 comma negative 5 right parenthesis

The equation of a circle with its center at the origin \((0, 0)\) is given by the standard form:

\[
x^2 + y^2 = r^2
\]

where \(r\) is the radius of the circle.

### Step 1: Use the distance formula to find \(r\)

The radius \(r\) is the distance between the center \((0, 0)\) and the point \((2, -5)\) on the circle. The distance formula is:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates of the center \((x_1, y_1) = (0, 0)\) and the point on the circle \((x_2, y_2) = (2, -5)\):

\[
r = \sqrt{(2 - 0)^2 + (-5 - 0)^2} = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}
\]

Thus, \(r = \sqrt{29}\).

### Step 2: Write the equation of the circle

Now that we know \(r^2 = 29\), the standard form of the equation of the circle is:

\[
x^2 + y^2 = 29
\]

This is the standard form of the equation of the circle.

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the equation change if the center were not at the origin?
2. Can you derive the general equation of a circle with center \((h, k)\) and radius \(r\)?
3. How would the radius change if the point were \((4, -6)\) instead of \((2, -5)\)?
4. What is the area of the circle in this problem?
5. How can you find the equation of a circle that is tangent to a specific line?

**Tip:** The square root of the sum of squared differences between coordinates gives the straight-line distance between two points.

Find the standard form of the equation of the circle having the following​ properties: Center at the origin Containing the point (2, -5)

Learn how to find the standard form of the equation of a circle with its center at the origin and containing the point (2, -5). This problem involves using the distance formula to calculate the radius and applying the standard circle equation. Suitable for students in Grades 9-12.

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