Point (-13, -7) On The Circle (x+13)^2 + (y+7)^2 = 25? A Detailed Analysis

Is the point (-13, -7) situated on the circle defined by the equation (x + 13)^2 + (y + 7)^2 = 25? This seemingly straightforward question delves into the fundamental relationship between points and circles in coordinate geometry. To definitively answer this, we must rigorously apply the definition of a circle and the equation that represents it. A circle, in its essence, is the locus of all points equidistant from a central point. This fixed distance is termed the radius, and the central point is, unsurprisingly, the center of the circle. The equation (x + 13)^2 + (y + 7)^2 = 25 is the standard form equation of a circle, which directly encodes the circle's center and radius. By carefully examining this equation, we can extract crucial information about the circle's characteristics. The standard form equation of a circle is generally expressed as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle and r denotes the radius. Comparing this general form to our specific equation, (x + 13)^2 + (y + 7)^2 = 25, we can immediately identify that the center of the circle is at the point (-13, -7). This is because the equation can be rewritten as (x - (-13))^2 + (y - (-7))^2 = 25, clearly showing that h = -13 and k = -7. Furthermore, the right-hand side of the equation, 25, represents the square of the radius (r^2). Therefore, to find the radius r, we take the square root of 25, which yields r = 5. This means that our circle has a radius of 5 units. Now that we have determined the center (-13, -7) and the radius 5 of the circle, we can proceed to verify whether the point (-13, -7) lies on the circle. To do this, we need to check if the coordinates of the point satisfy the equation of the circle. In other words, we will substitute x = -13 and y = -7 into the equation (x + 13)^2 + (y + 7)^2 = 25 and see if the equation holds true. This process of substitution is a fundamental technique in coordinate geometry, allowing us to determine the relationship between points and geometric figures. If the equation is satisfied, it means that the distance between the point and the center of the circle is equal to the radius, and thus the point lies on the circle. Conversely, if the equation is not satisfied, the point does not lie on the circle.

Verifying the Point (-13, -7)

To ascertain whether the point (-13, -7) resides on the circle defined by (x + 13)^2 + (y + 7)^2 = 25, we must meticulously substitute the x and y coordinates of the point into the equation. This process, a cornerstone of analytical geometry, effectively tests if the point adheres to the circle's defining condition: its distance from the center must equal the radius. Substituting x = -13 and y = -7 into the equation, we obtain: ((-13) + 13)^2 + ((-7) + 7)^2 = 25 This simplifies to: (0)^2 + (0)^2 = 25 Further simplification yields: 0 + 0 = 25 Which results in: 0 = 25 This final statement is patently false. The equation does not hold true when we substitute the coordinates of the point (-13, -7). This seemingly contradictory result reveals a crucial insight: the point (-13, -7) does not satisfy the equation of the circle. However, this outcome prompts a deeper reflection on our initial understanding of the circle and the point in question. We previously established that the center of the circle is indeed at the point (-13, -7). Therefore, how can a point coincide with the center of the circle and yet not lie on the circle itself? The resolution to this apparent paradox lies in the precise definition of a circle. A circle is the set of all points that are equidistant from the center, where that distance is the radius. The radius, in this case, is 5 units, as we calculated earlier. The point (-13, -7), being the center itself, has a distance of 0 from the center (itself). Since 0 is not equal to the radius 5, the point (-13, -7), while being the center, does not lie on the circle. It lies at the center of the circle. This subtle distinction is paramount in understanding the geometry of circles. Points on the circle are a specific distance (the radius) away from the center, while the center itself is not considered a point on the circle. This concept is crucial for accurately interpreting equations and geometric relationships involving circles. The act of substituting coordinates into equations and evaluating the resulting statements is a powerful tool in analytical geometry. It allows us to rigorously test hypotheses and uncover truths about geometric figures and their properties. In this case, the substitution revealed a fundamental characteristic of circles: the center is a distinct point, not included in the set of points that constitute the circle itself.

Distance Formula Confirmation

To further solidify our understanding and provide additional validation, we can employ the distance formula to calculate the distance between the point (-13, -7) and the center of the circle, which we know to be the same point, (-13, -7). The distance formula is a fundamental tool in coordinate geometry, derived from the Pythagorean theorem, that allows us to calculate the distance between any two points in a coordinate plane. The formula is expressed as: d = √((x₂ - x₁)² + (y₂ - y₁)²) where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points, and d represents the distance between them. In our case, we have the point (-13, -7) and the center of the circle, which is also (-13, -7). Let's designate (-13, -7) as (x₁, y₁) and (-13, -7) as (x₂, y₂). Substituting these values into the distance formula, we get: d = √((-13 - (-13))² + (-7 - (-7))²) Simplifying the expression inside the square root: d = √((0)² + (0)²) This further simplifies to: d = √(0 + 0) Which gives us: d = √0 Finally, we obtain: d = 0 The result of the distance formula calculation confirms that the distance between the point (-13, -7) and the center of the circle (-13, -7) is 0. This result is consistent with our previous finding that the point (-13, -7) does not lie on the circle because its distance from the center is not equal to the radius, which is 5. The distance formula provides a concrete mathematical verification of our earlier conclusion. The fact that the distance is 0 signifies that the point and the center are coincident; they occupy the same location in the coordinate plane. This reinforces the understanding that while the point (-13, -7) is the center of the circle, it is not a point on the circle itself. Points on the circle must be a distance of 5 units (the radius) away from the center. The application of the distance formula in this context highlights the interconnectedness of various concepts in coordinate geometry. We have seamlessly integrated the equation of a circle, the definition of a circle's center and radius, and the distance formula to arrive at a comprehensive and mathematically sound conclusion. This demonstrates the power of using multiple approaches to verify and deepen our understanding of geometric relationships.

Conclusion

In conclusion, through a rigorous application of the circle's equation and the distance formula, we have definitively determined that the point (-13, -7) does not lie on the circle defined by the equation (x + 13)^2 + (y + 7)^2 = 25. While the point (-13, -7) coincides with the center of the circle, it does not fulfill the condition of being a specific distance (the radius) away from the center, which is a requirement for a point to be considered on the circle. This exploration underscores the importance of understanding the precise definitions and properties of geometric figures. A circle is defined as the set of all points equidistant from a central point, and this distance is the radius. The center itself, while crucial in defining the circle, is not considered a point on the circle. This subtle distinction is paramount in accurately interpreting equations and geometric relationships. We utilized the standard form equation of a circle, (x - h)^2 + (y - k)^2 = r^2, to identify the center and radius. By substituting the coordinates of the point (-13, -7) into the equation, we demonstrated that the equation did not hold true, thus indicating that the point does not lie on the circle. Furthermore, we employed the distance formula, d = √((x₂ - x₁)² + (y₂ - y₁)²), to calculate the distance between the point and the center. The resulting distance of 0 confirmed that the point and the center are coincident, further reinforcing our conclusion. This comprehensive analysis highlights the power of analytical geometry in providing a rigorous and mathematical framework for understanding geometric concepts. By combining equations, formulas, and logical reasoning, we can precisely determine the relationships between points and geometric figures. The seemingly simple question of whether a point lies on a circle has led us to a deeper understanding of the fundamental properties of circles and the application of key concepts in coordinate geometry. The ability to analyze geometric problems using both algebraic and geometric methods is a crucial skill in mathematics, and this exploration serves as a valuable example of how these methods can be effectively integrated to arrive at a conclusive answer.