Circle Equation Explained Step-by-Step
Hey guys! Today, we're diving deep into the fascinating world of circles and their equations. We've got a fun problem on our hands, where we need to figure out the equation of a circle given the endpoints of its diameter. So, let's put on our thinking caps and get started!
The Challenge: Finding the Equation
Our mission, should we choose to accept it, is to find the equation of circle C. We know that the points J(-8,9) and K(-2,-5) mark the endpoints of a diameter of this circle. Armed with this information, we're going to embark on a journey to unveil the equation that perfectly represents this circle. The options laid out before us are:
A. (x-5)2+(y+2)2=58 B. (x-5)2+(y+2)2=232 C. (x+5)2+(y-2)2=58 D. (x+5)2+(y-2)2=232
Before we jump into calculations, let's take a moment to understand the fundamental concepts that govern the equation of a circle.
The Heart of the Circle Equation
The standard equation of a circle is like a secret code that reveals everything about a circle's identity. It's written as: (x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) is the center of the circle – the circle's heart, if you will.
- r is the radius – the distance from the center to any point on the circle's edge.
This equation is derived from the Pythagorean theorem, a cornerstone of geometry. Think of it as drawing a right triangle inside the circle, where the radius is the hypotenuse. The legs of this triangle are the horizontal and vertical distances from any point (x, y) on the circle to the center (h, k). This geometric relationship is beautifully captured by the equation. To successfully write the equation of our circle, we need to find two key pieces of information: the center (h, k) and the radius (r). Let's start by finding the circle's center.
Finding the Circle's Center: The Midpoint Magic
The center of the circle is like the captain of a ship, steering the entire shape. Since the diameter passes right through the center, the center is simply the midpoint of the diameter. Remember, we're given the endpoints of the diameter, J(-8, 9) and K(-2, -5). To find the midpoint, we use the midpoint formula, a handy tool that averages the x-coordinates and the y-coordinates of two points.
The midpoint formula is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Let's plug in our coordinates:
Midpoint = ((-8 + (-2))/2, (9 + (-5))/2) Midpoint = (-10/2, 4/2) Midpoint = (-5, 2)
Eureka! We've found the center of our circle. The center (h, k) is (-5, 2). This is a crucial step, guys, because it gives us two of the three pieces we need for our circle equation. Now, let's move on to finding the radius, the circle's vital measurement of size.
Determining the Radius: Measuring the Distance
The radius is the distance from the center of the circle to any point on its circumference. Since we know the center (-5, 2) and we also know the endpoints of the diameter, we can find the radius by calculating the distance between the center and one of the endpoints. We'll use the distance formula for this, which is another application of the Pythagorean theorem, but this time, we are calculating the length of a line segment given its endpoints.
The distance formula is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's use point J (-8, 9) and the center (-5, 2) to calculate the radius:
Radius = √((-8 - (-5))^2 + (9 - 2)^2) Radius = √((-3)^2 + (7)^2) Radius = √(9 + 49) Radius = √58
Awesome! We've found the radius, which is √58. Remember that the standard equation of a circle uses the radius squared (r^2), so we'll need to square this value later.
Crafting the Equation: Putting It All Together
Now comes the exciting part where we assemble all the pieces and write the equation of our circle. We have:
- Center (h, k) = (-5, 2)
- Radius (r) = √58
- Radius squared (r^2) = 58
Plugging these values into the standard equation of a circle, we get:
(x - (-5))^2 + (y - 2)^2 = 58
Simplifying, we have:
(x + 5)^2 + (y - 2)^2 = 58
Decoding the Options: Finding the Perfect Match
Let's revisit the options presented to us:
A. (x-5)2+(y+2)2=58 B. (x-5)2+(y+2)2=232 C. (x+5)2+(y-2)2=58 D. (x+5)2+(y-2)2=232
Comparing our derived equation, (x + 5)^2 + (y - 2)^2 = 58, with the options, we can clearly see that option C is the correct answer.
Why Option C Reigns Supreme
Option C, (x + 5)^2 + (y - 2)^2 = 58, perfectly matches the equation we derived. It correctly represents a circle with a center at (-5, 2) and a radius of √58. The other options either have the wrong center or the wrong radius squared.
- Options A and B have a center at (5, -2), which is incorrect.
- Option D has the correct center but the radius squared is 232, which doesn't match our calculation of 58.
Wrapping Up: Circle Equation Mastery
So, there you have it, guys! We've successfully navigated the world of circles and their equations. By understanding the standard equation, the midpoint formula, and the distance formula, we were able to find the equation of circle C. The correct answer is C. (x+5)2+(y-2)2=58. Remember, practice makes perfect, so keep exploring those geometric concepts and tackling those problems. Until next time, keep circling around those challenging questions!
