Finding The Equation Of A Circle With Center (-2, 10) And Point (10, 5)

Hey guys! Today, we're diving deep into the fascinating world of circles and their equations. We've got a classic problem on our hands, and we're going to break it down step-by-step, making sure everyone understands the nitty-gritty details. Our mission? To find the equation of circle C, given its center and a point it passes through. So, let's put on our math hats and get started!

The Circle's Tale: Center and a Point

Our mathematical adventure begins with the knowledge that circle C has its center nestled at the coordinates (-2, 10). Imagine that point as the heart of our circle, the epicenter from which all points on the circumference are equidistant. We also know that this circle isn't a loner; it embraces the point P(10, 5) on its journey. This point is like a friendly landmark on the circle's edge, giving us crucial information about its size and position. Our ultimate quest is to translate these clues into a precise equation that represents circle C. This equation will be our mathematical fingerprint, uniquely identifying this circle among all others in the coordinate plane.

Decoding the Circle Equation: The Core Formula

Before we jump into the specifics of our problem, let's take a moment to appreciate the general form of a circle's equation. This equation is the key to unlocking the mysteries of circles, and it's rooted in the Pythagorean theorem, a fundamental concept in geometry. The equation is elegantly expressed as:

$$(x - h)^2 + (y - k)^2 = r^2$$

Where:

• (h, k) bravely represents the coordinates of the circle's center.
• r stands tall as the radius, the distance from the center to any point on the circle's edge.
• (x, y) symbolizes any point that dares to reside on the circle's circumference.

This equation is a powerful statement about the relationship between the center, radius, and all the points that make up the circle. It tells us that for any point (x, y) on the circle, the sum of the squared horizontal distance (x - h) and the squared vertical distance (y - k) from the center is always equal to the squared radius. This is the essence of a circle, captured in a neat algebraic expression.

Finding the Missing Piece: Calculating the Radius

Now that we've befriended the general equation of a circle, let's roll up our sleeves and apply it to our specific problem. We know the center of circle C is at (-2, 10), and it graciously passes through the point P(10, 5). The missing link in our equation is the radius, r. Fear not, for we have the tools to unearth this vital piece of information!

The radius, as we know, is the distance from the center of the circle to any point on its circumference. Since we have the center (-2, 10) and a point on the circle (10, 5), we can employ the trusty distance formula to calculate this distance. The distance formula, a direct descendant of the Pythagorean theorem, is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Where:

• (x1, y1) and (x2, y2) are the coordinates of our two points.
• d is the distance between them.

Let's plug in our coordinates and watch the magic happen:

$$r = \sqrt{(10 - (-2))^2 + (5 - 10)^2}$$

$$r = \sqrt{(12)^2 + (-5)^2}$$

$$r = \sqrt{144 + 25}$$

$$r = \sqrt{169}$$

$$r = 13$$

Eureka! We've discovered that the radius of circle C is a cheerful 13 units. This value is the key to unlocking the final equation of our circle.

Crafting the Equation: Putting It All Together

With the radius in our grasp and the center coordinates firmly in mind, we're ready to assemble the equation of circle C. It's like piecing together a puzzle, where each piece fits perfectly to reveal the complete picture. We'll revisit our general equation:

$$(x - h)^2 + (y - k)^2 = r^2$$

And substitute the values we've unearthed:

• h = -2 (the x-coordinate of the center)
• k = 10 (the y-coordinate of the center)
• r = 13 (the radius)

Plugging these values into the equation, we get:

$$(x - (-2))^2 + (y - 10)^2 = 13^2$$

$$(x + 2)^2 + (y - 10)^2 = 169$$

Voila! This, my friends, is the equation that represents circle C. It's a compact and elegant expression that encapsulates all the essential information about our circle: its center, its radius, and the relationship between its points.

The Final Flourish: Selecting the Correct Answer

Now that we've derived the equation of circle C, let's take a look at the options presented to us and select the one that matches our result. We're looking for:

$$(x + 2)^2 + (y - 10)^2 = 169$$

Comparing this equation to the options, we can confidently choose the correct answer.

Wrapping Up Our Circle Expedition

Great job, everyone! We've successfully navigated the world of circles, deciphered their equations, and emerged victorious. We started with a circle's center and a point, calculated its radius, and crafted its unique equation. This journey has reinforced our understanding of the fundamental relationship between a circle's geometry and its algebraic representation.

Remember, the equation of a circle is more than just a formula; it's a powerful tool that allows us to describe and analyze these fascinating shapes. Keep exploring, keep questioning, and keep unraveling the mysteries of mathematics!

Main keywords: circle equation, radius calculation, distance formula

Additional Practice Problems

To solidify your understanding, here are some additional practice problems:

1. A circle has a center at (3, -2) and passes through the point (7, 1). Find the equation of the circle.
2. The diameter of a circle has endpoints at (-1, 4) and (5, 2). Determine the equation of the circle.
3. A circle is tangent to the x-axis and has a center at (2, 3). Write the equation of the circle.

These problems will challenge you to apply the concepts we've discussed in different scenarios. Remember to break each problem down into smaller steps, identify the given information, and choose the appropriate formulas. With practice, you'll become a circle equation master in no time!

Further Exploration

If you're eager to delve even deeper into the world of circles, here are some topics you might find fascinating:

• Conic Sections: Circles are part of a larger family of curves known as conic sections. Exploring ellipses, parabolas, and hyperbolas will broaden your geometric horizons.
• Circles in Coordinate Geometry: Investigate how circles interact with lines and other geometric figures in the coordinate plane. You'll discover some elegant and surprising relationships.
• Applications of Circles: Circles appear in countless real-world applications, from gears and wheels to planetary orbits and radio waves. Exploring these applications will highlight the power and versatility of this fundamental shape.

So, keep your curiosity burning, and never stop exploring the wonders of mathematics! Remember, every problem is an opportunity to learn and grow. Until next time, happy calculating!