Identifying Points On A Circle (x+7)^2+(y-10)^2=13^2

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In the realm of mathematics, circles hold a fundamental place, and understanding their equations is crucial for various applications. This article delves into the intricacies of identifying points that lie on a circle, focusing on the equation (x+7)2+(y-10)2=13^2. We will explore the underlying concepts, provide a step-by-step solution to the given problem, and offer a broader perspective on circle equations and their properties.

Understanding the Circle Equation

Before diving into the specifics, let's first grasp the fundamental equation of a circle. The standard form of a circle's equation is given by:

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

Where:

  • (h, k) represents the coordinates of the circle's center.
  • r denotes the radius of the circle.

This equation stems directly from the Pythagorean theorem, which relates the sides of a right triangle. Imagine a circle centered at (h, k). Any point (x, y) on the circle forms the hypotenuse of a right triangle, with the legs being the horizontal distance (x - h) and the vertical distance (y - k) from the center. The radius 'r' is the length of this hypotenuse. Applying the Pythagorean theorem, we get the circle equation.

Now, let's connect this to our specific problem. The equation (x+7)2+(y-10)2=13^2 represents a circle with:

  • Center: (-7, 10)
  • Radius: 13

This is because (x + 7) is equivalent to (x - (-7)), and the square root of 13^2 is 13. The center's coordinates are the values that, when plugged into the equation, make the squared terms zero.

Determining Points on the Circle

To determine if a point lies on the circle, we simply substitute the point's coordinates (x, y) into the circle's equation. If the equation holds true (i.e., the left-hand side equals the right-hand side), then the point lies on the circle. If the equation does not hold true, the point does not lie on the circle.

This process essentially checks if the distance between the point and the circle's center is equal to the radius. If the distance is equal to the radius, the point is on the circle. If the distance is less than the radius, the point is inside the circle. If the distance is greater than the radius, the point is outside the circle.

Let's illustrate this with a simple example. Consider a circle with the equation x^2 + y^2 = 25. This circle has a center at (0, 0) and a radius of 5. To check if the point (3, 4) lies on this circle, we substitute x = 3 and y = 4 into the equation:

3^2 + 4^2 = 9 + 16 = 25

Since 25 equals the right-hand side of the equation, the point (3, 4) lies on the circle.

Solving the Problem: A Step-by-Step Approach

Now, let's tackle the problem at hand. We need to determine which of the given points lies on the circle represented by the equation (x+7)2+(y-10)2=13^2. We will do this by substituting the coordinates of each point into the equation and checking if the equation holds true.

A. (5, 12)

Substitute x = 5 and y = 12 into the equation:

(5 + 7)^2 + (12 - 10)^2 = 12^2 + 2^2 = 144 + 4 = 148

Since 148 ≠ 13^2 (which is 169), the point (5, 12) does not lie on the circle.

B. (-7, -3)

Substitute x = -7 and y = -3 into the equation:

(-7 + 7)^2 + (-3 - 10)^2 = 0^2 + (-13)^2 = 0 + 169 = 169

Since 169 = 13^2, the point (-7, -3) lies on the circle.

C. (-6, -10)

Substitute x = -6 and y = -10 into the equation:

(-6 + 7)^2 + (-10 - 10)^2 = 1^2 + (-20)^2 = 1 + 400 = 401

Since 401 ≠ 13^2, the point (-6, -10) does not lie on the circle.

D. (6, 23)

Substitute x = 6 and y = 23 into the equation:

(6 + 7)^2 + (23 - 10)^2 = 13^2 + 13^2 = 169 + 169 = 338

Since 338 ≠ 13^2, the point (6, 23) does not lie on the circle.

Therefore, after evaluating all the options, we can confidently conclude that the point B. (-7, -3) is the only point that lies on the circle represented by the equation (x+7)2+(y-10)2=13^2.

Visualizing the Solution

To further solidify our understanding, it's helpful to visualize the circle and the points in question. Imagine a coordinate plane. The circle is centered at (-7, 10) and has a radius of 13 units. The point (-7, -3) lies directly below the center, 13 units away, thus confirming it lies on the circle. The other points, when plotted, would be located either inside or outside the circle.

Graphing the circle and the points provides a visual representation of the solution and can aid in comprehension, especially for those who are more visually inclined. Tools like graphing calculators or online graphing utilities can be used to create such visualizations.

Beyond the Basics: Applications of Circle Equations

The equation of a circle isn't just a mathematical abstraction; it has numerous real-world applications. Here are a few examples:

  • Navigation: Circles are used in navigation systems to determine distances and bearings. For instance, GPS systems rely on the intersection of circles to pinpoint a location.
  • Engineering: In engineering, circles are fundamental in designing circular structures like wheels, pipes, and tunnels. The equation of a circle helps engineers calculate dimensions and ensure structural integrity.
  • Computer Graphics: Circles are essential in computer graphics for creating smooth curves and shapes. They are used in rendering software and game development to create realistic visuals.
  • Astronomy: Circles (or ellipses, which are stretched circles) are used to model the orbits of planets and other celestial bodies. Understanding the equations of these shapes is crucial for astronomical calculations.

These are just a few examples, and the applications of circle equations extend to many other fields, highlighting the importance of understanding this fundamental mathematical concept.

Common Mistakes and How to Avoid Them

When working with circle equations, several common mistakes can occur. Being aware of these pitfalls can help prevent errors and improve accuracy.

  • Incorrectly Identifying the Center: A common mistake is misinterpreting the signs in the equation. Remember that the center is (h, k), where h and k are the values subtracted from x and y, respectively. For example, in the equation (x + 7)^2 + (y - 10)^2 = 13^2, the center is (-7, 10), not (7, -10).
  • Forgetting to Square the Radius: Another mistake is forgetting to take the square root of the right-hand side of the equation to find the radius. For instance, if the equation is (x - h)^2 + (y - k)^2 = 25, the radius is √25 = 5, not 25.
  • Arithmetic Errors: Careless arithmetic errors during substitution and calculation can lead to incorrect answers. Double-check your calculations to ensure accuracy.
  • Misunderstanding the Concept: A fundamental misunderstanding of the circle equation and its relationship to the Pythagorean theorem can lead to confusion. Make sure you have a solid grasp of the underlying concepts.

By being mindful of these common mistakes and practicing regularly, you can avoid errors and develop a strong understanding of circle equations.

Practice Problems

To further enhance your understanding, try solving these practice problems:

  1. Does the point (2, -1) lie on the circle (x - 2)^2 + (y + 1)^2 = 9?
  2. Find the center and radius of the circle represented by the equation (x + 5)^2 + y^2 = 16.
  3. Write the equation of a circle with center (1, -3) and radius 4.

Working through these problems will help you solidify your knowledge and build confidence in your ability to work with circle equations.

Conclusion

In conclusion, determining whether a point lies on a circle involves substituting the point's coordinates into the circle's equation and verifying if the equation holds true. We successfully identified that the point (-7, -3) lies on the circle represented by (x+7)2+(y-10)2=13^2. Furthermore, we explored the broader context of circle equations, their applications, common mistakes to avoid, and provided practice problems for further learning. Mastering circle equations is a valuable skill in mathematics, with applications spanning various fields. By understanding the underlying concepts and practicing regularly, you can confidently tackle problems involving circles and their equations.

This exploration of circle equations not only provides a solution to the specific problem but also offers a comprehensive understanding of the topic, equipping you with the knowledge and skills to tackle similar challenges in the future. Remember, the key to success in mathematics lies in understanding the fundamentals, practicing regularly, and visualizing the concepts.