Circle Equation: Point (-5,-3), Center (-2,1)

Understanding the equation of a circle is fundamental in coordinate geometry. A circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the center of the circle, and the fixed distance is called the radius. The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is given by: $(x - h)^2 + (y - k)^2 = r^2$. This equation is derived directly from the distance formula, which calculates the distance between two points in a Cartesian coordinate system. If we consider any point $(x, y)$ on the circle and the center $(h, k)$, the distance between them is always equal to the radius $r$. Applying the distance formula, we get $\sqrt{(x-h)^2 + (y-k)^2} = r$. Squaring both sides of this equation leads us to the standard form we use today: $(x - h)^2 + (y - k)^2 = r^2$. This equation is incredibly useful because it allows us to quickly identify the center and radius of a circle just by looking at its algebraic representation. For instance, if we have the equation $(x-3)^2 + (y+4)^2 = 16$, we can immediately tell that the center is at $(3, -4)$ and the radius is $\sqrt{16} = 4$. Conversely, if we know the center and radius, we can construct the equation of the circle. This problem asks us to find the specific equation of a circle given its center and a point that lies on its circumference. This means we have enough information to determine both the center $(h, k)$ and the radius $r$, and thus write its equation. The key is to use the given point to calculate the radius. The radius is the distance between the center and any point on the circle. By calculating this distance, we can find $r^2$ and complete the equation.

To solve this problem, we are given that the center of the circle is at $(-2, 1)$ and that the circle contains the point $(-5, -3)$. Using the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, we can substitute the coordinates of the center $(h, k) = (-2, 1)$ into the equation. This gives us $(x - (-2))^2 + (y - 1)^2 = r^2$, which simplifies to $(x + 2)^2 + (y - 1)^2 = r^2$. Now, we need to find the value of $r^2$. Since the point $(-5, -3)$ lies on the circle, it must satisfy the circle's equation. We can substitute $x = -5$ and $y = -3$ into the equation we have so far to solve for $r^2$. So, we have $(-5 + 2)^2 + (-3 - 1)^2 = r^2$. Let's calculate the terms: $(-5 + 2) = -3$, and $(-3 - 1) = -4$. Squaring these values, we get $(-3)^2 = 9$ and $(-4)^2 = 16$. Adding these squared values together, we find $r^2 = 9 + 16 = 25$. Therefore, the equation of the circle that contains the point $(-5, -3)$ and has a center at $(-2, 1)$ is $(x + 2)^2 + (y - 1)^2 = 25$. This equation perfectly encapsulates all the properties of the circle defined by the given information. It's a direct application of the standard circle equation, where the distance from the center to any point on the circumference is constant and equal to the radius. The process involves identifying the components of the center $(h, k)$ and then using the given point $(x, y)$ to determine the radius squared, $r^2$. This methodical approach ensures accuracy and clarity in deriving the final equation. The derived equation $(x + 2)^2 + (y - 1)^2 = 25$ is the correct representation, and we can now compare it with the given options to select the right answer. The structure of the equation clearly shows the center at $(-2, 1)$ and the squared radius as $25$. This allows us to easily verify our result.

Now, let's analyze the given options to find the one that matches our derived equation. The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. We are given the center $(h, k) = (-2, 1)$ and a point on the circle $(-5, -3)$. First, let's correctly incorporate the center into the equation. For the x-coordinate of the center, $h = -2$, so $(x - h)$ becomes $(x - (-2))$, which simplifies to $(x + 2)$. For the y-coordinate of the center, $k = 1$, so $(y - k)$ becomes $(y - 1)$. Thus, the left side of our equation is $(x + 2)^2 + (y - 1)^2$. Next, we need to determine the value of $r^2$. The radius $r$ is the distance between the center $(-2, 1)$ and the point on the circle $(-5, -3)$. We can use the distance formula, $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, or simply substitute the point into our partially formed equation. Using the latter method, we substitute $x = -5$ and $y = -3$ into $(x + 2)^2 + (y - 1)^2 = r^2$:

$(-5 + 2)^2 + (-3 - 1)^2 = r^2$

Calculate the terms inside the parentheses:

$(-3)^2 + (-4)^2 = r^2$

Square the numbers:

$9 + 16 = r^2$

Add the results:

$25 = r^2$

So, the equation of the circle is $(x + 2)^2 + (y - 1)^2 = 25$. Now, let's examine the provided options:

• A. $(x-1)^2+(y+2)^2=25$: This equation represents a circle with center $(1, -2)$ and radius $5$. This does not match our center.
• B. $(x+2)^2+(y-1)^2=5$: This equation has the correct center $(-2, 1)$, but the radius squared is $5$, meaning the radius is $\sqrt{5}$. This does not match our calculated $r^2 = 25$.
• C. $(x+2)^2+(y-1)^2=25$: This equation has the center $(-2, 1)$ and radius squared $25$. This perfectly matches our derived equation.
• D. $(x-1)^2+(y+2)^2=5$: This equation represents a circle with center $(1, -2)$ and radius squared $5$. This does not match our center or radius.

Therefore, the correct equation representing the circle that contains the point $(-5, -3)$ and has a center at $(-2, 1)$ is Option C. This solution confirms our understanding of how to construct and interpret the equation of a circle using its center and a point on its circumference. The key takeaway is the systematic application of the standard circle equation and the distance formula (or its implicit use in substitution) to find the unknown radius squared.

Understanding the Components of a Circle's Equation

The equation of a circle in its standard form, $(x - h)^2 + (y - k)^2 = r^2$, is a powerful tool in analytical geometry. Each part of this equation conveys specific information about the circle's position and size. The variables $h$ and $k$ represent the coordinates of the center of the circle. Specifically, the center is located at the point $(h, k)$. It's important to note the signs: if the equation has $(x - h)^2$, the x-coordinate of the center is positive $h$. If it has $(x + h)^2$, which is equivalent to $(x - (-h))^2$, then the x-coordinate of the center is negative $h$. The same logic applies to the $y$ term and the $k$ value. The term $r^2$ on the right side of the equation represents the square of the radius of the circle. The radius, $r$, is the distance from the center to any point on the circle's circumference. Therefore, if you know $r^2$, you can easily find the radius by taking the square root: $r = \sqrt{r^2}$. This standard form is derived from the Pythagorean theorem and the distance formula. Imagine a right-angled triangle formed by the horizontal distance $(x - h)$, the vertical distance $(y - k)$, and the radius $r$ as the hypotenuse. According to the Pythagorean theorem, $(x - h)^2 + (y - k)^2 = r^2$. This relationship holds true for every point $(x, y)$ on the circle.

In the problem we are addressing, we are given the center $(h, k) = (-2, 1)$ and a point on the circle $(-5, -3)$. To find the equation of the circle, we first substitute the center's coordinates into the standard form. This yields $(x - (-2))^2 + (y - 1)^2 = r^2$, which simplifies to $(x + 2)^2 + (y - 1)^2 = r^2$. The next crucial step is to determine the value of $r^2$. Since the point $(-5, -3)$ lies on the circle, its coordinates must satisfy the circle's equation. We can find $r^2$ by substituting $x = -5$ and $y = -3$ into the equation:

$(-5 + 2)^2 + (-3 - 1)^2 = r^2$

Calculating the differences within the parentheses gives us:

$(-3)^2 + (-4)^2 = r^2$

Squaring these values, we get:

$9 + 16 = r^2$

Adding these results provides the value of $r^2$:

$25 = r^2$

With $r^2 = 25$, we can now write the complete equation of the circle: $(x + 2)^2 + (y - 1)^2 = 25$. This equation accurately describes a circle centered at $(-2, 1)$ with a radius of $5$ (since $r = \sqrt{25} = 5$). Comparing this to the options provided, we see that Option C is the correct match. This detailed breakdown reinforces how the standard equation of a circle relates geometric properties (center and radius) to an algebraic expression, enabling us to solve problems involving circles in the coordinate plane. The ability to manipulate and interpret these equations is a cornerstone of advanced mathematics and its applications.

Verifying the Solution with the Distance Formula

To further solidify our understanding and verify the equation of the circle, let's explicitly use the distance formula to calculate the radius. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. In our case, the two points are the center of the circle $(h, k) = (-2, 1)$ and the point on the circle $(x, y) = (-5, -3)$. The distance between these two points is the radius, $r$. Let's plug in the values:

$r = \sqrt{(-5 - (-2))^2 + (-3 - 1)^2}$

Simplify the terms inside the square root:

$r = \sqrt{(-5 + 2)^2 + (-3 - 1)^2}$

$r = \sqrt{(-3)^2 + (-4)^2}$

Now, square the differences:

$r = \sqrt{9 + 16}$

Add the squared values:

$r = \sqrt{25}$

Calculate the square root to find the radius:

$r = 5$

Since the standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, we need the value of $r^2$. Squaring our calculated radius, $r = 5$, we get $r^2 = 5^2 = 25$. Now, we substitute the coordinates of the center $(h, k) = (-2, 1)$ into the standard equation:

$(x - (-2))^2 + (y - 1)^2 = 5^2$

$(x + 2)^2 + (y - 1)^2 = 25$

This result perfectly matches Option C. This verification process using the distance formula confirms that our derived equation is correct. It highlights the fundamental relationship between the definition of a circle (a set of points equidistant from a center) and its algebraic representation in the coordinate plane. The distance formula is essentially the basis for the circle equation, and using it directly provides a robust way to check our answers. This consistent outcome across different approaches (substitution into the partial equation versus explicit use of the distance formula) builds confidence in our understanding of circle equations. It's a demonstration of how mathematical concepts are interconnected and reinforce each other, leading to reliable solutions.

Conclusion

In summary, to find the equation of a circle given its center and a point it passes through, we utilize the standard form $(x - h)^2 + (y - k)^2 = r^2$. We substitute the coordinates of the center $(h, k)$ into the equation and then use the coordinates of the given point $(x, y)$ to calculate the value of $r^2$. For this specific problem, the center is $(-2, 1)$ and the point is $(-5, -3)$. Substituting the center gives us $(x + 2)^2 + (y - 1)^2 = r^2$. Plugging in the point $(-5, -3)$ allows us to calculate $r^2 = (-5 + 2)^2 + (-3 - 1)^2 = (-3)^2 + (-4)^2 = 9 + 16 = 25$. Thus, the final equation is $(x + 2)^2 + (y - 1)^2 = 25$. This corresponds to Option C. Understanding the standard equation of a circle is key to solving problems involving circles in coordinate geometry. For further exploration of conic sections and their equations, you can visit **

mathworld.wolfram.com.**