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.