Equation Of A Circle Centered At (3,5) Passing Through (-2,9)

The equation of a circle is a fundamental concept in coordinate geometry, playing a crucial role in various mathematical and real-world applications. This article aims to provide a comprehensive understanding of how to determine the equation of a circle given its center and a point it passes through. Specifically, we will focus on finding the equation of a circle centered at $(3,5)$ and passing through the point $(-2,9)$. This involves applying the standard equation of a circle and utilizing the distance formula to calculate the radius. By the end of this article, you will be equipped with the knowledge and skills to solve similar problems effectively.

Standard Equation of a Circle

To begin, let’s discuss the standard equation of a circle. This equation is the foundation for determining the circle’s properties and representation in the coordinate plane. The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is given by:

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

In this equation:

• $(x, y)$ represents any point on the circle.
• $(h, k)$ represents the coordinates of the center of the circle.
• $r$ represents the radius of the circle, which is the distance from the center to any point on the circle.

This equation is derived from the Pythagorean theorem and provides a straightforward way to describe a circle's location and size. The equation essentially states that for any point $(x, y)$ on the circle, the square of the horizontal distance from $x$ to the center's x-coordinate ($h$) plus the square of the vertical distance from $y$ to the center's y-coordinate ($k$) is equal to the square of the radius ($r$). This relationship holds true for all points on the circle, thus defining its shape and size uniquely.

Applying the Standard Equation

When we are given the center and a point on the circle, we can use this standard equation to find the specific equation for that circle. The process involves substituting the coordinates of the center $(h, k)$ into the equation and then determining the radius $r$. If the radius is not directly given, it can be calculated using the distance formula between the center and the given point on the circle. Once the radius is found, we can substitute its value into the standard equation to obtain the complete equation of the circle.

For example, if we know the center is at $(3, 5)$, we can substitute $h = 3$ and $k = 5$ into the standard equation, which gives us:

$$(x - 3)^2 + (y - 5)^2 = r^2$$

To find the radius $r$, we need additional information, such as a point on the circle. This point will allow us to calculate the distance from the center to the point, which is the radius. Once we find the radius, we can square it and substitute it into the equation to complete the equation of the circle. Understanding and applying the standard equation is essential for solving problems related to circles in coordinate geometry, making it a cornerstone concept for further mathematical studies.

Determining the Radius

To determine the specific equation of the circle centered at $(3,5)$ and passing through the point $(-2,9)$, the crucial step is to find the radius ($r$). The radius is the distance between the center of the circle and any point on its circumference. In this case, we are given the center $(3,5)$ and a point on the circle $(-2,9)$. We can use the distance formula to calculate the radius.

The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points in a coordinate plane. If we have two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is given by:

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

In our problem, the center of the circle is $(3,5)$, which we can denote as $(x_1, y_1)$, and the point on the circle is $(-2,9)$, which we can denote as $(x_2, y_2)$. Plugging these coordinates into the distance formula, we get:

$$r = \sqrt{(-2 - 3)^2 + (9 - 5)^2}$$

$$r = \sqrt{(-5)^2 + (4)^2}$$

$$r = \sqrt{25 + 16}$$

$$r = \sqrt{41}$$

Thus, the radius of the circle is $\sqrt{41}$. This value represents the distance from the center $(3,5)$ to the point $(-2,9)$ on the circle. Now that we have found the radius, we can proceed to determine the equation of the circle.

Significance of the Radius

The radius is a fundamental property of a circle that determines its size. Knowing the radius allows us to fully define the circle's equation and understand its dimensions. In the context of the equation of a circle, the radius appears squared ($r^2$), which represents the square of the distance from the center to any point on the circle. This squared value is what we will use in the standard equation of the circle.

In our case, since the radius $r$ is $\sqrt{41}$, the square of the radius $r^2$ is $(\sqrt{41})^2 = 41$. This value will be used on the right-hand side of the circle's equation. Understanding how to calculate and interpret the radius is essential for solving problems involving circles in coordinate geometry. The distance formula provides a direct method for finding the radius when given the center and a point on the circle, making it a crucial tool in our problem-solving process. Now that we have determined the radius squared, we can substitute this value into the standard equation to find the equation of the circle.

Forming the Equation of the Circle

Having found the radius $r = \sqrt{41}$, we can now form the equation of the circle. We know the standard equation of a circle with center $(h, k)$ and radius $r$ is:

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

In our case, the center of the circle is $(3, 5)$, so $h = 3$ and $k = 5$. We have also calculated that $r^2 = 41$. Substituting these values into the standard equation, we get:

$$(x - 3)^2 + (y - 5)^2 = 41$$

This equation represents the circle centered at $(3, 5)$ and passing through the point $(-2, 9)$. It defines the relationship between the x and y coordinates of any point on the circle. The left side of the equation represents the sum of the squares of the horizontal and vertical distances from any point $(x, y)$ on the circle to the center $(3, 5)$, and the right side is the square of the radius.

Understanding the Equation

The equation $(x - 3)^2 + (y - 5)^2 = 41$ encapsulates all the information about the circle. The terms $(x - 3)^2$ and $(y - 5)^2$ represent the squared horizontal and vertical distances, respectively, from a point $(x, y)$ on the circle to the center $(3, 5)$. The sum of these squared distances is always equal to the square of the radius, which is 41 in this case.

This equation allows us to verify whether a point lies on the circle. If we substitute the coordinates of a point into the equation and the equation holds true, then the point lies on the circle. For example, if we substitute the point $(-2, 9)$ into the equation, we get:

$$(-2 - 3)^2 + (9 - 5)^2 = (-5)^2 + (4)^2 = 25 + 16 = 41$$

Since this is equal to the right side of the equation, the point $(-2, 9)$ indeed lies on the circle. The equation provides a concise and complete representation of the circle, enabling us to analyze its properties and solve related problems. By understanding the components of the equation, we can readily identify the circle's center and radius, and use this information to further explore its characteristics and relationships with other geometric figures.

Analyzing the Answer Choices

Now that we have derived the equation of the circle, $(x - 3)^2 + (y - 5)^2 = 41$, we can analyze the given answer choices to determine which one matches our result. The answer choices are:

• A. $(x - 3)^2 + (y - 5)^2 = 41$
• B. $(x - 3)^2 + (y - 5)^2 = 17$
• C. $(x + 3)^2 + (y + 5)^2 = 17$
• D. $(x + 3)^2 + (y + 5)^2 = 41$

Comparing our derived equation with the answer choices, we can see that:

• Option A, $(x - 3)^2 + (y - 5)^2 = 41$, exactly matches the equation we found. This indicates that option A is the correct answer.
• Option B, $(x - 3)^2 + (y - 5)^2 = 17$, has the correct center $(3, 5)$, but the radius squared is 17, which does not match our calculated value of 41. Thus, option B is incorrect.
• Option C, $(x + 3)^2 + (y + 5)^2 = 17$, represents a circle with center $(-3, -5)$, which is different from the given center $(3, 5)$. Also, the radius squared is 17, which is incorrect. Therefore, option C is incorrect.
• Option D, $(x + 3)^2 + (y + 5)^2 = 41$, also represents a circle with center $(-3, -5)$, which is different from the given center $(3, 5)$. Although the radius squared is 41, the incorrect center makes option D incorrect.

Conclusion on Answer Choices

From this analysis, it is clear that option A, $(x - 3)^2 + (y - 5)^2 = 41$, is the correct equation representing a circle centered at $(3, 5)$ and passing through the point $(-2, 9)$. The other options either have the wrong center or the wrong radius, making them incorrect. This systematic approach of deriving the equation and then comparing it with the given choices is a reliable method for solving problems involving circles in coordinate geometry. By understanding the standard equation of a circle and how to calculate the radius, we can confidently determine the correct answer.

In conclusion, finding the equation of a circle centered at $(3,5)$ and passing through the point $(-2,9)$ involves understanding and applying the standard equation of a circle and the distance formula. By substituting the center coordinates into the standard equation and calculating the radius using the distance formula, we determined that the equation of the circle is $(x - 3)^2 + (y - 5)^2 = 41$. This equation accurately represents the circle with the given properties.

This process highlights the importance of understanding fundamental concepts in coordinate geometry, such as the standard equation of a circle and the distance formula. These tools enable us to solve a variety of problems involving circles, lines, and other geometric figures in the coordinate plane. The ability to apply these concepts effectively is crucial for success in mathematics and related fields.

By systematically working through the problem, we were able to derive the correct equation and verify it against the given answer choices. This methodical approach ensures accuracy and provides a clear understanding of the underlying principles. The equation of a circle is a powerful tool in mathematics, with applications ranging from basic geometry to advanced calculus and beyond. Mastering the concepts discussed in this article will undoubtedly enhance your problem-solving skills and deepen your understanding of mathematical principles.