Finding the Equation of a Circle Given the Ends of a Diameter

Given the coordinates of the endpoints of a diameter, how do we find the equation of the corresponding circle? Understanding this process helps in various applications in geometry, and it aligns perfectly with the optimization strategies Google appreciates for SEO. This article covers the step-by-step methodology, providing a comprehensive guide for students and professionals.

Understanding the Problem

The coordinates of the endpoints of the diameter of a circle are given as (2, 3) and (5, -6). Our goal is to find the equation of the circle. To achieve this, we need to follow several steps, including finding the center of the circle and the radius.

Step 1: Finding the Center of the Circle

The center of the circle is the midpoint of the diameter. The formula for the midpoint (M) between two points (x_1, y_1) and (x_2, y_2) is:

M left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)

For the given points (2, 3) and (5, -6), we compute the midpoint as follows:

M left(frac{2 5}{2}, frac{3 - 6}{2}right) left(frac{7}{2}, frac{-3}{2}right) (3.5, -1.5)

Step 2: Calculating the Radius

The radius of the circle is half the distance between the endpoints of the diameter. To find this distance, we use the distance formula:

(x_2 - x_1)^2 (y_2 - y_1)^2

For our points, we have:

d sqrt{(5 - 2)^2 (-6 - 3)^2} sqrt{3^2 (-9)^2} sqrt{9 81} sqrt{90} 3sqrt{10}

The radius (r) is then:

r frac{d}{2} frac{3sqrt{10}}{2}

Step 3: Writing the Equation of the Circle

The standard form of the equation of a circle with center ((h, k)) and radius (r) is:

(x - h)^2 (y - k)^2 r^2

Substituting (h 3.5), (k -1.5), and (r frac{3sqrt{10}}{2}), we get:

(x - 3.5)^2 (y 1.5)^2 left(frac{3sqrt{10}}{2}right)^2

Simplifying the right-hand side:

left(frac{3sqrt{10}}{2}right)^2 frac{9 cdot 10}{4} frac{90}{4} frac{45}{2}

Thus, the equation of the circle is:

(x - 3.5)^2 (y 1.5)^2 frac{45}{2}

Simplified Equation

The given problem also states that the equation of the circle can be expressed as:

x^2 - 2x - 5y 3y^2 - 6 0

While this form is provided, expanding and simplifying it to the standard form gives:

x^2 - 7x 49/4 y^2 3y 9/4 - 45/2 0

Further simplification results in:

x^2 y^2 - 7x 3y - 8 0

Conclusion

The equation of the circle corresponding to the given diameter endpoints is ( (x - 3.5)^2 (y 1.5)^2 frac{45}{2} ). This circle is centered at (3.5, -1.5) with a radius of ( frac{3sqrt{10}}{2} ).

Understanding how to find the equation of a circle from the ends of its diameter is crucial for many applications, from real-world geometry problems to more complex mathematical scenarios. Practicing with similar problems ensures a solid grasp of the underlying principles.