Finding The Circle Equation Standard Form With Diameter Endpoints

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#Points P(-10, 10) and Q(6, -2) form the diameter of a circle. Let's embark on a journey to determine the equation of this circle in its standard form. This exploration will not only provide the answer but also illuminate the underlying principles of circles and their equations.

Understanding the Standard Form of a Circle's Equation

Before diving into the specifics of this problem, let's first establish a firm understanding of the standard form equation of a circle. The standard form equation of a circle is expressed as:

(x - h)² + (y - k)² = r²

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r denotes the radius of the circle.

This equation encapsulates the fundamental properties of a circle: its center and its radius. By knowing these two parameters, we can precisely define a circle in the coordinate plane. The equation stems from the Pythagorean theorem, applied to the distance between any point (x, y) on the circle and the center (h, k). This distance, by definition, is the radius 'r'.

Finding the Circle's Center: The Midpoint Formula

The points P(-10, 10) and Q(6, -2) define the diameter of our circle. The diameter is a line segment that passes through the center of the circle and has endpoints on the circle's circumference. Crucially, the center of the circle is the midpoint of this diameter. To find the midpoint, we employ the midpoint formula:

Midpoint ((x₁, y₁), (x₂, y₂)) = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Applying this formula to our points P(-10, 10) and Q(6, -2), we get:

Center (h, k) = ((-10 + 6)/2, (10 + (-2))/2) = (-4/2, 8/2) = (-2, 4)

Therefore, the center of our circle is located at the point (-2, 4). This point is equidistant from both P and Q, lying precisely in the middle of the diameter. The coordinates (-2, 4) will be our (h, k) values in the standard form equation.

Calculating the Circle's Radius: The Distance Formula

Now that we've pinpointed the center of the circle, our next step is to determine its radius. The radius is the distance from the center of the circle to any point on its circumference. Since we know the center (-2, 4) and two points on the circle, P(-10, 10) and Q(6, -2), we can use the distance formula to calculate the radius. The distance formula is given by:

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

We can use either point P or point Q to calculate the radius. Let's use point P(-10, 10) and the center (-2, 4):

Radius (r) = √((-10 - (-2))² + (10 - 4)²) = √((-8)² + (6)²) = √(64 + 36) = √100 = 10

Alternatively, we can use point Q(6, -2) and the center (-2, 4):

Radius (r) = √((6 - (-2))² + (-2 - 4)²) = √((8)² + (-6)²) = √(64 + 36) = √100 = 10

As expected, both calculations yield the same result: the radius of the circle is 10 units. This confirms the consistency of our calculations and reinforces our understanding of the circle's geometry. Now we have all the pieces needed to construct the equation of the circle.

Constructing the Circle's Equation in Standard Form

With the center (h, k) = (-2, 4) and the radius r = 10 determined, we can now assemble the equation of the circle in its standard form:

(x - h)² + (y - k)² = r²

Substituting the values we found, we get:

(x - (-2))² + (y - 4)² = 10²

Simplifying this equation, we arrive at:

(x + 2)² + (y - 4)² = 100

This is the equation of the circle in standard form. It clearly displays the circle's center (-2, 4) and its radius of 10 units. This equation represents all the points (x, y) that lie on the circumference of the circle. Any point that satisfies this equation will be exactly 10 units away from the center (-2, 4).

Identifying the Correct Option

Comparing our derived equation, (x + 2)² + (y - 4)² = 100, with the given options, we find that it matches option D:

D. (x + 2)² + (y - 4)² = 100

Therefore, option D is the correct answer. This equation accurately describes the circle defined by the diameter with endpoints P(-10, 10) and Q(6, -2).

Alternative Approach: Diameter and Radius Relationship

Another way to approach this problem is to first find the length of the diameter using the distance formula and then halve it to get the radius. Let's calculate the distance between points P(-10, 10) and Q(6, -2):

Diameter = √((6 - (-10))² + (-2 - 10)²) = √((16)² + (-12)²) = √(256 + 144) = √400 = 20

The diameter is 20 units, so the radius is half of that, which is 10 units. This confirms our previous calculation of the radius. This method provides an alternative pathway to finding the radius, reinforcing the concept of the relationship between diameter and radius.

Key Concepts Revisited

Let's reiterate the key concepts involved in solving this problem:

  1. Standard Form of a Circle's Equation: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  2. Midpoint Formula: Used to find the center of the circle given the endpoints of the diameter.
  3. Distance Formula: Used to calculate the distance between two points, which can be used to find the radius or the diameter.
  4. Diameter and Radius Relationship: The radius of a circle is half the length of its diameter.

By mastering these concepts, you can confidently tackle a wide range of problems involving circles and their equations. These concepts are fundamental in geometry and have applications in various fields, including engineering, physics, and computer graphics.

Common Mistakes to Avoid

When solving problems involving circles, it's crucial to avoid common pitfalls. Here are a few mistakes to watch out for:

  1. Incorrectly Applying the Midpoint Formula: Ensure you add the x-coordinates and y-coordinates separately before dividing by 2.
  2. Misusing the Distance Formula: Double-check your signs and ensure you're subtracting the coordinates in the correct order.
  3. Confusing Diameter and Radius: Remember that the radius is half the diameter. Using the diameter as the radius in the equation will lead to an incorrect answer.
  4. Sign Errors in the Standard Form: Pay close attention to the signs in the standard form equation. The center's coordinates are subtracted from x and y, so a center of (-2, 4) will result in (x + 2) and (y - 4) in the equation.

By being aware of these potential errors, you can increase your accuracy and confidence in solving circle-related problems. Practice and attention to detail are key to mastering these concepts.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. The endpoints of the diameter of a circle are A(2, 5) and B(-4, -1). Find the equation of the circle in standard form.
  2. A circle has a center at (3, -2) and a radius of 5. Write the equation of the circle in standard form.
  3. The equation of a circle is (x - 1)² + (y + 3)² = 16. Determine the center and radius of the circle.

Working through these problems will reinforce your understanding of the concepts and techniques discussed in this guide. Remember to apply the midpoint formula, distance formula, and the standard form equation of a circle to arrive at the solutions. Check your answers carefully and review the steps if needed.

Conclusion

In this comprehensive guide, we've meticulously determined the equation of the circle with diameter endpoints P(-10, 10) and Q(6, -2) to be (x + 2)² + (y - 4)² = 100. We achieved this by first finding the center of the circle using the midpoint formula and then calculating the radius using the distance formula. We then substituted these values into the standard form equation of a circle.

This journey has not only provided the answer to the problem but has also deepened our understanding of circles, their equations, and the fundamental geometric principles that govern them. By mastering these concepts and avoiding common mistakes, you can confidently navigate a wide array of circle-related problems in mathematics and beyond. Keep practicing, keep exploring, and keep expanding your mathematical horizons!