Finding the Point that Lies on the Circle

Oct 13, 2025 by miodeya.com 133 views

Understanding the Basics of a Circle

A circle is a set of points that are all equidistant from a central point known as the center. The distance from the center to any point on the circle is called the radius. In this problem, we are given the radius of the circle as 5 and the center point as (-1,4). We need to find the point that lies on the circle.

The Equation of a Circle

The equation of a circle with center (h,k) and radius r is given by:

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

In this case, the center of the circle is (-1,4) and the radius is 5. So, the equation of the circle is:

(x + 1)^2 + (y - 4)^2 = 25

Finding the Point that Lies on the Circle

To find the point that lies on the circle, we need to substitute the coordinates of each option into the equation of the circle and check if it satisfies the equation.

Option A: (3,-5)

Substituting the coordinates of option A into the equation of the circle, we get:

(3 + 1)^2 + (-5 - 4)^2 = 4^2 + (-9)^2 = 16 + 81 = 97

Since 97 is not equal to 25, option A does not lie on the circle.

Option B: (5,7)

Substituting the coordinates of option B into the equation of the circle, we get:

(5 + 1)^2 + (7 - 4)^2 = 6^2 + 3^2 = 36 + 9 = 45

Since 45 is not equal to 25, option B does not lie on the circle.

Option C: (-4,8)

Substituting the coordinates of option C into the equation of the circle, we get:

(-4 + 1)^2 + (8 - 4)^2 = (-3)^2 + 4^2 = 9 + 16 = 25

Since 25 is equal to 25, option C lies on the circle.

Option D: (-1,6)

Substituting the coordinates of option D into the equation of the circle, we get:

(-1 + 1)^2 + (6 - 4)^2 = 0^2 + 2^2 = 0 + 4 = 4

Since 4 is not equal to 25, option D does not lie on the circle.

Conclusion

Based on the calculations above, we can conclude that the point that lies on the circle is option C: (-4,8).

Key Takeaways

The equation of a circle with center (h,k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2.
To find the point that lies on the circle, we need to substitute the coordinates of each option into the equation of the circle and check if it satisfies the equation.
The point that lies on the circle is option C: (-4,8).

Real-World Applications

Understanding the equation of a circle and how to find the point that lies on the circle has many real-world applications, such as:

Geometry and Trigonometry: The equation of a circle is used to solve problems in geometry and trigonometry.
Computer Graphics: The equation of a circle is used to create 2D and 3D graphics.
Engineering: The equation of a circle is used to design and analyze circular structures such as bridges and tunnels.

Common Mistakes

Not using the correct equation of the circle: Make sure to use the correct equation of the circle, which is (x - h)^2 + (y - k)^2 = r^2.
Not substituting the coordinates correctly: Make sure to substitute the coordinates of each option into the equation of the circle correctly.
Not checking if the equation is satisfied: Make sure to check if the equation is satisfied for each option.

Tips and Tricks

Use the equation of the circle to solve problems: The equation of the circle is a powerful tool that can be used to solve many problems in mathematics and real-world applications.
Practice, practice, practice: The more you practice using the equation of the circle, the more comfortable you will become with it.
Use online resources: There are many online resources available that can help you learn and practice using the equation of the circle.

Q: What is the equation of a circle?

A: The equation of a circle with center (h,k) and radius r is given by:

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

Q: How do I find the point that lies on the circle?

A: To find the point that lies on the circle, you need to substitute the coordinates of each option into the equation of the circle and check if it satisfies the equation.

Q: What is the center of the circle?

A: The center of the circle is the point from which the distance to any point on the circle is equal to the radius. In the equation of the circle, the center is represented by (h,k).

Q: What is the radius of the circle?

A: The radius of the circle is the distance from the center to any point on the circle. In the equation of the circle, the radius is represented by r.

Q: How do I determine if a point lies on the circle?

A: To determine if a point lies on the circle, you need to substitute the coordinates of the point into the equation of the circle and check if it satisfies the equation.

Q: What is the significance of the equation of the circle?

A: The equation of the circle is a powerful tool that can be used to solve many problems in mathematics and real-world applications. It is used to find the point that lies on the circle, determine the center and radius of the circle, and solve problems in geometry and trigonometry.

Q: Can I use the equation of the circle to solve problems in other fields?

A: Yes, the equation of the circle can be used to solve problems in other fields such as computer graphics, engineering, and physics.

Q: How do I use the equation of the circle to solve problems in computer graphics?

A: In computer graphics, the equation of the circle is used to create 2D and 3D graphics. You can use the equation of the circle to create circles, ellipses, and other shapes.

Q: How do I use the equation of the circle to solve problems in engineering?

A: In engineering, the equation of the circle is used to design and analyze circular structures such as bridges and tunnels. You can use the equation of the circle to determine the center and radius of the circle, and to solve problems in geometry and trigonometry.

Q: How do I use the equation of the circle to solve problems in physics?

A: In physics, the equation of the circle is used to describe the motion of objects in circular paths. You can use the equation of the circle to determine the center and radius of the circle, and to solve problems in kinematics and dynamics.

Q: What are some common mistakes to avoid when using the equation of the circle?

A: Some common mistakes to avoid when using the equation of the circle include:

Not using the correct equation of the circle
Not substituting the coordinates correctly
Not checking if the equation is satisfied
Not considering the center and radius of the circle

Q: How can I practice using the equation of the circle?

A: You can practice using the equation of the circle by solving problems and exercises in mathematics and real-world applications. You can also use online resources and software to practice using the equation of the circle.

Q: What are some real-world applications of the equation of the circle?

A: Some real-world applications of the equation of the circle include:

Computer graphics
Engineering
Physics
Geometry and trigonometry
Design and analysis of circular structures

Q: How can I use the equation of the circle to solve problems in my field?

A: You can use the equation of the circle to solve problems in your field by applying the concepts and techniques learned in mathematics and real-world applications. You can also use online resources and software to practice using the equation of the circle.

Q: What are some tips and tricks for using the equation of the circle?

A: Some tips and tricks for using the equation of the circle include:

Using the correct equation of the circle
Substituting the coordinates correctly
Checking if the equation is satisfied
Considering the center and radius of the circle
Practicing using the equation of the circle regularly

Q: How can I learn more about the equation of the circle?

A: You can learn more about the equation of the circle by reading books and articles, taking online courses and tutorials, and practicing using the equation of the circle. You can also use online resources and software to learn more about the equation of the circle.