Identifying Points On A Circle The Equation (x-3)^2+(y+4)^2=6^2 Explained

In mathematics, especially in analytic geometry, understanding the equation of a circle is crucial. The standard form equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ is the radius. This equation stems from the Pythagorean theorem and describes all the points $(x, y)$ that lie on the circumference of the circle. The given problem asks us to identify which point, among the provided options, lies on the circle represented by the equation $(x - 3)^2 + (y + 4)^2 = 6^2$. To solve this, we need to substitute the coordinates of each point into the equation and check if the equation holds true. This involves a straightforward application of algebraic principles, where we plug in the $x$ and $y$ values of each point and simplify to see if the result equals $6^2$, which is 36. The point that satisfies this condition is the one that lies on the circle. Let's delve deeper into the step-by-step process of verifying each option, highlighting the mathematical concepts involved and why each correct or incorrect choice leads to a specific conclusion.

Step-by-Step Verification of Points

The essence of determining whether a point lies on a circle is to verify if its coordinates satisfy the circle's equation. This involves substituting the $x$ and $y$ coordinates of the point into the equation $(x - 3)^2 + (y + 4)^2 = 6^2$ and checking if the left-hand side equals the right-hand side (36). Let's systematically evaluate each of the provided options:

Option A: (9, -2)

To check if the point $(9, -2)$ lies on the circle, we substitute $x = 9$ and $y = -2$ into the equation:

$(9 - 3)^2 + (-2 + 4)^2 = 6^2 + 2^2 = 36 + 4 = 40$

Since $40 e 36$, the point $(9, -2)$ does not lie on the circle. The result of 40 indicates that this point is located outside the circle, as its distance from the center is greater than the radius.

Option B: (0, 11)

Next, we test the point $(0, 11)$ by substituting $x = 0$ and $y = 11$ into the equation:

$(0 - 3)^2 + (11 + 4)^2 = (-3)^2 + (15)^2 = 9 + 225 = 234$

Since $234 e 36$, the point $(0, 11)$ also does not lie on the circle. The large value of 234 indicates that this point is significantly far from the circle's circumference, much farther than the radius would allow.

Option C: (3, 10)

Now, we substitute $x = 3$ and $y = 10$ for the point $(3, 10)$:

$(3 - 3)^2 + (10 + 4)^2 = 0^2 + (14)^2 = 0 + 196 = 196$

As $196 e 36$, the point $(3, 10)$ is not on the circle. This point, similar to option B, is located well outside the circle due to the large result after substitution.

Option D: (-9, 4)

For the point $(-9, 4)$, we substitute $x = -9$ and $y = 4$:

$(-9 - 3)^2 + (4 + 4)^2 = (-12)^2 + (8)^2 = 144 + 64 = 208$

Since $208 e 36$, the point $(-9, 4)$ does not lie on the circle either. This high value confirms that the point is far removed from the circle's boundary.

Option E: (-3, -4)

Finally, we check the point $(-3, -4)$ by substituting $x = -3$ and $y = -4$:

$(-3 - 3)^2 + (-4 + 4)^2 = (-6)^2 + (0)^2 = 36 + 0 = 36$

Since $36 = 36$, the point $(-3, -4)$ lies on the circle. This is the correct answer because the coordinates satisfy the equation of the circle, indicating it is a point on the circumference.

Detailed Explanation of the Correct Answer

The correct answer is E. (-3, -4). This point satisfies the equation $(x - 3)^2 + (y + 4)^2 = 6^2$ because when we substitute $x = -3$ and $y = -4$, we get:

$(-3 - 3)^2 + (-4 + 4)^2 = (-6)^2 + (0)^2 = 36 + 0 = 36$

This result confirms that the point $(-3, -4)$ is indeed on the circle. Understanding why this is the only correct answer involves appreciating the geometric interpretation of the circle's equation. The equation represents all points that are exactly 6 units away from the center of the circle, which is $(3, -4)$. The point $(-3, -4)$ is precisely 6 units away from $(3, -4)$, satisfying this condition. The other points, when substituted, yield values greater than 36, indicating they are farther than 6 units from the center and thus lie outside the circle. The process of verifying each point underscores the importance of accurately applying the distance formula, derived from the Pythagorean theorem, in the context of circles and their equations.

Common Mistakes and How to Avoid Them

When solving problems involving circle equations, several common mistakes can lead to incorrect answers. Recognizing these pitfalls and understanding how to avoid them is crucial for achieving accuracy. Here are some frequent errors and strategies to prevent them:

1. Incorrect Substitution: A very common mistake is substituting the coordinates incorrectly into the equation. For example, students might mix up the x and y values or not apply the correct signs. To avoid this, always double-check the substitution, writing each step clearly and deliberately. Pay close attention to the signs in the equation and the coordinates of the point.

2. Arithmetic Errors: Mistakes in arithmetic, such as squaring numbers or adding values incorrectly, can lead to wrong results. To minimize these errors, perform calculations carefully, possibly using a calculator for complex computations. Double-check each step of your arithmetic to ensure accuracy.

3. Misunderstanding the Equation: A fundamental misunderstanding of the circle equation can lead to incorrect solutions. Students may not correctly identify the center and radius from the equation. To prevent this, ensure a solid understanding of the standard form equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Practice identifying the center and radius from various circle equations.

4. Forgetting to Square the Radius: Another common mistake is forgetting to square the radius when checking if a point lies on the circle. The equation equals $r^2$, not $r$. To avoid this, always remember to square the radius when you are verifying points or determining the circle's properties.

5. Not Checking All Options: In multiple-choice questions, students might stop after finding one point that seems to fit, without verifying that it's the only correct answer. To ensure you select the right answer, check all options, especially when multiple options seem plausible. This practice helps confirm your solution and can catch any errors made in earlier attempts.

By being aware of these common mistakes and actively working to avoid them, students can improve their accuracy and confidence in solving circle equation problems.

Conclusion: Mastering Circle Equations

In conclusion, determining whether a point lies on a circle involves a straightforward yet crucial process of substituting the point's coordinates into the circle's equation and verifying if the equation holds true. The correct answer in this case, E. (-3, -4), demonstrates this principle effectively. By meticulously substituting the coordinates of each given point into the equation $(x - 3)^2 + (y + 4)^2 = 6^2$, we systematically eliminated the incorrect options and identified the one point that satisfies the equation. This method underscores the importance of accurate algebraic manipulation and a solid understanding of the circle's equation. The common mistakes discussed, such as incorrect substitution, arithmetic errors, and misunderstanding the equation, serve as valuable lessons for students to avoid similar pitfalls in the future. Mastering circle equations is not just about finding the right answer; it's about developing a deeper comprehension of geometric principles and enhancing problem-solving skills. This understanding extends beyond mere equation-solving and encompasses the ability to visualize geometric concepts, apply algebraic techniques, and interpret results meaningfully. As students continue to practice and apply these skills, they will build a strong foundation in analytic geometry and related mathematical fields. The ability to confidently and accurately solve problems involving circles is a testament to one's mathematical proficiency and lays the groundwork for tackling more complex challenges in higher mathematics.

By understanding the underlying principles and practicing diligently, students can confidently tackle problems involving circles and their equations.