Circle Centered At Origin Finding Points On Circle

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In the realm of geometry, circles hold a fundamental place, and understanding their properties is crucial for solving various mathematical problems. This article delves into a specific scenario involving a circle centered at the origin and aims to identify points that lie on its circumference. We will explore the concept of the equation of a circle, the distance formula, and how to apply these principles to determine whether a given point lies on a circle. Let's embark on this geometric journey to unravel the solution.

Problem Statement

Consider a circle denoted as $C$, which is centered at the origin (0, 0). A point $Q$ with coordinates (10, 0) is known to lie on this circle $C$. The objective is to identify which of the following points also lie on circle $C$:

A. $(3, \sqrt{91})$ B. $(4, 5\sqrt{3})$ C. $(5, 5\sqrt{3})$ D. $(6, 4)$

Understanding the Equation of a Circle

To effectively solve this problem, we need to understand the equation of a circle. A circle is defined as the set of all points equidistant from a central point. This distance is known as the radius of the circle. When the circle is centered at the origin (0, 0), the equation of the circle takes a simplified form:

$x^2 + y^2 = r^2$

where:

• $(x, y)$ represents any point on the circle.
• $r$ represents the radius of the circle.

This equation stems directly from the Pythagorean theorem, which relates the sides of a right triangle. Imagine a point $(x, y)$ on the circle. The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance from the origin. These two distances form the legs of a right triangle, and the radius of the circle acts as the hypotenuse. The Pythagorean theorem then states that:

$x^2 + y^2 = r^2$

This equation is fundamental to solving our problem. By determining the radius of the circle, we can then check whether the given points satisfy this equation, thus indicating whether they lie on the circle.

Determining the Radius

We are given that the point $Q(10, 0)$ lies on the circle $C$. This provides us with crucial information to determine the radius of the circle. Since $Q$ lies on the circle, its coordinates must satisfy the equation of the circle:

$x^2 + y^2 = r^2$

Substituting the coordinates of $Q(10, 0)$ into the equation, we get:

$10^2 + 0^2 = r^2$

$100 + 0 = r^2$

$100 = r^2$

Taking the square root of both sides, we obtain the radius:

$r = \sqrt{100} = 10$

Therefore, the radius of circle $C$ is 10 units. Now that we know the radius, we can write the complete equation of circle $C$:

$x^2 + y^2 = 10^2$

$x^2 + y^2 = 100$

This equation will serve as our primary tool to check which of the given points lie on the circle.

Verifying the Points

Now that we have the equation of the circle, $x^2 + y^2 = 100$, we can substitute the coordinates of each given point into the equation and check if the equation holds true. If the equation is satisfied, then the point lies on the circle; otherwise, it does not.

A. $(3, \sqrt{91})$

Substitute $x = 3$ and $y = \sqrt{91}$ into the equation:

$3^2 + (\sqrt{91})^2 = 9 + 91 = 100$

Since the equation is satisfied, the point $(3, \sqrt{91})$ lies on circle $C$.

B. $(4, 5\sqrt{3})$

Substitute $x = 4$ and $y = 5\sqrt{3}$ into the equation:

$4^2 + (5\sqrt{3})^2 = 16 + 25 * 3 = 16 + 75 = 91$

Since 91 is not equal to 100, the point $(4, 5\sqrt{3})$ does not lie on circle $C$.

C. $(5, 5\sqrt{3})$

Substitute $x = 5$ and $y = 5\sqrt{3}$ into the equation:

$5^2 + (5\sqrt{3})^2 = 25 + 25 * 3 = 25 + 75 = 100$

Since the equation is satisfied, the point $(5, 5\sqrt{3})$ lies on circle $C$.

D. $(6, 4)$

Substitute $x = 6$ and $y = 4$ into the equation:

$6^2 + 4^2 = 36 + 16 = 52$

Since 52 is not equal to 100, the point $(6, 4)$ does not lie on circle $C$.

Conclusion

By systematically applying the equation of a circle and verifying each point, we have determined that the points $(3, \sqrt{91})$ and $(5, 5\sqrt{3})$ lie on circle $C$. This problem demonstrates the fundamental principles of coordinate geometry and the application of algebraic equations to solve geometric problems. Understanding the equation of a circle and the distance formula are essential skills for tackling such challenges. This process underscores the interconnectedness of algebra and geometry in mathematical problem-solving.

This article explores the concepts of a circle centered at the origin and how to determine if specific points lie on that circle. We will walk through the process of finding the radius of a circle given a point on its circumference and then use this information to verify if other points are also part of the circle. This mathematical problem is grounded in coordinate geometry, blending algebraic equations with geometric principles.

Problem Restatement

Imagine a circle, which we'll call $C$, perfectly centered at the origin (the point where the x and y axes intersect, represented as (0,0)). We know that a specific point, $Q$, with coordinates (10, 0), resides directly on the circumference of this circle. The challenge is to identify which of the following points also lie on circle $C$:

A. $(3, \sqrt{91})$ B. $(4, 5\sqrt{3})$ C. $(5, 5\sqrt{3})$ D. $(6, 4)$

To solve this, we'll need to dive into the fundamental properties of circles and how they're represented mathematically.

The Foundation: The Equation of a Circle

The key to understanding circles in a coordinate plane is their equation. A circle is defined as a collection of points that are all the same distance away from a central point. This distance is, of course, the radius of the circle. When the circle's center is conveniently located at the origin (0, 0), the equation becomes wonderfully simple:

$x^2 + y^2 = r^2$

Let's break this down:

• $(x, y)$ represents the coordinates of any point that lies on the circle's edge.
• $r$ stands for the radius of the circle – the constant distance from the center to any point on the circle.

This equation is a direct application of the Pythagorean Theorem, a cornerstone of geometry. Think of it this way: For any point $(x, y)$ on the circle, we can visualize a right triangle. The horizontal leg has a length of $x$, the vertical leg has a length of $y$, and the hypotenuse (the distance from the origin to the point) is the radius, $r$. The Pythagorean Theorem then states:

$x^2 + y^2 = r^2$

This equation is our powerful tool. If we know the radius, we can plug in the coordinates of any point and see if they “fit” the equation. If they do, the point lies on the circle; if not, it doesn't.

Unveiling the Radius

Our problem gives us a valuable clue: the point $Q(10, 0)$ is on the circle. Since Q lies on the circle, its coordinates must satisfy the circle's equation. Let's substitute these values into our equation:

$x^2 + y^2 = r^2$

$(10)^2 + (0)^2 = r^2$

$100 + 0 = r^2$

$100 = r^2$

To find the radius, $r$, we simply take the square root of both sides:

$r = \sqrt{100} = 10$

So, the radius of circle $C$ is 10 units. Now we can write the complete equation for this specific circle:

$x^2 + y^2 = 10^2$

$x^2 + y^2 = 100$

This is the equation we'll use to test the candidate points.

The Moment of Truth: Verifying the Points

With our equation, $x^2 + y^2 = 100$, we can now systematically check each of the provided points. We'll substitute the x and y coordinates of each point into the equation. If the equation holds true (equals 100), then the point lies on the circle. Let's get started.

A. $(3, \sqrt{91})$ – Does it Fit?

Let's substitute $x = 3$ and $y = \sqrt{91}$ into the equation:

$(3)^2 + (\sqrt{91})^2 = 9 + 91 = 100$

Hallelujah! The equation is satisfied. This means the point $(3, \sqrt{91})$ does indeed lie on circle $C$.

B. $(4, 5\sqrt{3})$ – Another Candidate

Now, let's try $x = 4$ and $y = 5\sqrt{3}$:

$(4)^2 + (5\sqrt{3})^2 = 16 + (25 * 3) = 16 + 75 = 91$

Alas, 91 is not equal to 100. The point $(4, 5\sqrt{3})$ does not lie on the circle.

C. $(5, 5\sqrt{3})$ – On to the Next

Let's test $x = 5$ and $y = 5\sqrt{3}$:

$(5)^2 + (5\sqrt{3})^2 = 25 + (25 * 3) = 25 + 75 = 100$

Success! This equation holds true, so the point $(5, 5\sqrt{3})$ is also on circle $C$.

D. $(6, 4)$ – The Final Check

Finally, let's substitute $x = 6$ and $y = 4$:

$(6)^2 + (4)^2 = 36 + 16 = 52$

Sadly, 52 is not equal to 100. Therefore, the point $(6, 4)$ is not on the circle.

Wrapping It Up: The Verdict

By systematically applying the circle's equation and checking each point's coordinates, we've successfully identified which points lie on circle $C$. Our analysis reveals that the points $(3, \sqrt{91})$ and $(5, 5\sqrt{3})$ are the residents of this circle.

This exercise beautifully demonstrates the power of coordinate geometry. We've combined algebraic equations with geometric concepts to solve a concrete problem. Understanding the equation of a circle, how it relates to the Pythagorean Theorem, and how to apply it is a crucial skill in mathematics. This method highlights the elegant way in which algebra and geometry work together to describe and analyze the world around us. By understanding the relationship between a circle's equation and the distance of points from its center, we can solve geometric problems effectively. The circle equation gives a clear connection between coordinates and the constant radius, showcasing math's logical beauty.

In mathematics, particularly in the study of geometry, understanding the properties of circles is fundamental. This article presents a problem centered around identifying points that lie on a circle centered at the origin. We will walk through the process, emphasizing the importance of the circle's equation and the use of the Pythagorean theorem in solving geometric problems. This comprehensive guide aims to provide clarity and a step-by-step approach to similar mathematical challenges.

The Problem Unveiled

Consider a circle, denoted by $C$, which is perfectly centered at the origin, the point (0, 0) on the coordinate plane. We are given that point $Q$, with coordinates (10, 0), lies directly on the circle's circumference. The challenge is to determine which of the following points also lie on circle $C$:

A. $(3, \sqrt{91})$ B. $(4, 5\sqrt{3})$ C. $(5, 5\sqrt{3})$ D. $(6, 4)$

To tackle this problem, we will leverage the equation of a circle and verify each point individually.

Core Concept: The Equation of a Circle

The equation of a circle is the cornerstone of our solution. A circle is defined as the set of all points that are equidistant from a central point. This distance is known as the radius. When the circle is centered at the origin, the equation simplifies to:

$x^2 + y^2 = r^2$

Here's what each component signifies:

• $(x, y)$ represents the coordinates of any point on the circle's circumference.
• $r$ is the radius of the circle, the constant distance from the center to any point on the circle.

This equation is a direct manifestation of the Pythagorean theorem. Imagine a right triangle formed by the x-coordinate, the y-coordinate, and the radius as the hypotenuse. The theorem states that the sum of the squares of the two legs (x and y) is equal to the square of the hypotenuse (r), thus $x^2 + y^2 = r^2$. This equation is crucial for determining if a point lies on the circle.

Finding the Circle's Radius

We are given that point $Q(10, 0)$ lies on the circle. This information allows us to calculate the radius. Since $Q$ is on the circle, its coordinates must satisfy the circle's equation:

$x^2 + y^2 = r^2$

Substituting the coordinates of $Q$ into the equation, we have:

$(10)^2 + (0)^2 = r^2$

$100 + 0 = r^2$

$100 = r^2$

Taking the square root of both sides gives us:

$r = \sqrt{100} = 10$

Thus, the radius of circle $C$ is 10 units. Now, we can express the complete equation of circle $C$ as:

$x^2 + y^2 = 10^2$

$x^2 + y^2 = 100$

This equation will serve as our benchmark for testing the given points.

Testing the Points: Verification Process

With the equation of circle $C$ established as $x^2 + y^2 = 100$, we can now substitute the coordinates of each given point into the equation. If the equation holds true, the point lies on the circle; otherwise, it does not. Let's proceed with each point.

A. The Point $(3, \sqrt{91})$

Substitute $x = 3$ and $y = \sqrt{91}$ into the equation:

$(3)^2 + (\sqrt{91})^2 = 9 + 91 = 100$

Since the equation is satisfied, the point $(3, \sqrt{91})$ lies on circle $C$.

B. Analyzing $(4, 5\sqrt{3})$

Substitute $x = 4$ and $y = 5\sqrt{3}$ into the equation:

$(4)^2 + (5\sqrt{3})^2 = 16 + (25 * 3) = 16 + 75 = 91$

Since 91 ≠ 100, the point $(4, 5\sqrt{3})$ does not lie on circle $C$.

C. Investigating $(5, 5\sqrt{3})$

Substitute $x = 5$ and $y = 5\sqrt{3}$ into the equation:

$(5)^2 + (5\sqrt{3})^2 = 25 + (25 * 3) = 25 + 75 = 100$

The equation is satisfied, so the point $(5, 5\sqrt{3})$ lies on circle $C$.

D. Evaluating $(6, 4)$

Substitute $x = 6$ and $y = 4$ into the equation:

$(6)^2 + (4)^2 = 36 + 16 = 52$

Since 52 ≠ 100, the point $(6, 4)$ does not lie on circle $C$.

Conclusion: Identifying Points on the Circle

Through systematic substitution and verification, we have determined that the points $(3, \sqrt{91})$ and $(5, 5\sqrt{3})$ lie on circle $C$. This problem demonstrates the practical application of the circle's equation in coordinate geometry. Understanding and applying this equation, derived from the Pythagorean theorem, is crucial in solving a variety of geometric problems.

This process underscores the power of algebra in geometry, where equations provide a precise tool for analyzing geometric figures. By calculating the radius and applying the circle equation, we efficiently identified the points on the circle. Math’s interconnectedness and its use in solving real-world problems are beautifully demonstrated in this problem. The circle's symmetrical beauty and the algebraic precision in locating points are key takeaways.