This Circle Is Centered At The Origin, And The Length Of Its Radius Is 6. What Is The Circle's Equation?A. X + Y = 36 X+y=36 X + Y = 36 B. X 6 + Y 6 = 1 X^6+y^6=1 X 6 + Y 6 = 1 C. X 2 + Y 2 = 36 X^2+y^2=36 X 2 + Y 2 = 36

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In mathematics, 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. When the center of the circle is at the origin (0, 0) and the radius is 6, we can use this information to find the equation of the circle.

The General Equation of a Circle

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

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

Applying the General Equation to the Given Circle

In this case, the center of the circle is at the origin (0, 0), so h = 0 and k = 0. The radius of the circle is 6, so r = 6. Substituting these values into the general equation, we get:

(x - 0)^2 + (y - 0)^2 = 6^2

Simplifying the equation, we get:

x^2 + y^2 = 36

Comparing the Derived Equation with the Options

Now that we have derived the equation of the circle, let's compare it with the given options:

A. x + y = 36 B. x^6 + y^6 = 1 C. x^2 + y^2 = 36

The derived equation matches option C, which is x^2 + y^2 = 36.

Conclusion

In this article, we have learned how to find the equation of a circle with a given center and radius. We applied the general equation of a circle to the given problem and derived the equation x^2 + y^2 = 36. This equation matches option C, which is the correct answer.

Key Takeaways

  • The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
  • To find the equation of a circle with center (h, k) and radius r, substitute the values of h, k, and r into the general equation.
  • The equation of a circle with center (0, 0) and radius 6 is x^2 + y^2 = 36.

Frequently Asked Questions

Q: What is the general equation of a circle? A: The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.

Q: How do I find the equation of a circle with a given center and radius? A: Substitute the values of h, k, and r into the general equation.

Understanding Circle Equations

In mathematics, 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. When the center of the circle is at the origin (0, 0) and the radius is 6, we can use this information to find the equation of the circle.

The General Equation of a Circle

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

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

Applying the General Equation to the Given Circle

In this case, the center of the circle is at the origin (0, 0), so h = 0 and k = 0. The radius of the circle is 6, so r = 6. Substituting these values into the general equation, we get:

(x - 0)^2 + (y - 0)^2 = 6^2

Simplifying the equation, we get:

x^2 + y^2 = 36

Comparing the Derived Equation with the Options

Now that we have derived the equation of the circle, let's compare it with the given options:

A. x + y = 36 B. x^6 + y^6 = 1 C. x^2 + y^2 = 36

The derived equation matches option C, which is x^2 + y^2 = 36.

Conclusion

In this article, we have learned how to find the equation of a circle with a given center and radius. We applied the general equation of a circle to the given problem and derived the equation x^2 + y^2 = 36. This equation matches option C, which is the correct answer.

Key Takeaways

  • The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
  • To find the equation of a circle with center (h, k) and radius r, substitute the values of h, k, and r into the general equation.
  • The equation of a circle with center (0, 0) and radius 6 is x^2 + y^2 = 36.

Frequently Asked Questions

Q&A Section

Q: What is the general equation of a circle? A: The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.

Q: How do I find the equation of a circle with a given center and radius? A: Substitute the values of h, k, and r into the general equation.

Q: What is the equation of a circle with center (0, 0) and radius 6? A: The equation of a circle with center (0, 0) and radius 6 is x^2 + y^2 = 36.

Q: What is the difference between the equation of a circle and the equation of an ellipse? A: The equation of a circle is (x - h)^2 + (y - k)^2 = r^2, while the equation of an ellipse is ((x - h)2/a2) + ((y - k)2/b2) = 1.

Q: Can I use the general equation of a circle to find the equation of a circle with a center at (3, 4) and radius 5? A: Yes, you can use the general equation of a circle to find the equation of a circle with a center at (3, 4) and radius 5. Substitute the values of h = 3, k = 4, and r = 5 into the general equation.

Q: What is the equation of a circle with center (0, 0) and radius 0? A: The equation of a circle with center (0, 0) and radius 0 is x^2 + y^2 = 0.

Q: Can I use the general equation of a circle to find the equation of a circle with a center at (0, 0) and an infinite radius? A: No, you cannot use the general equation of a circle to find the equation of a circle with a center at (0, 0) and an infinite radius. The general equation of a circle requires a finite radius.

Q: What is the equation of a circle with center (0, 0) and a negative radius? A: The equation of a circle with center (0, 0) and a negative radius is not defined. The radius of a circle must be a non-negative number.

Conclusion

In this article, we have answered some of the most frequently asked questions about circle equations. We have learned how to find the equation of a circle with a given center and radius, and we have compared the derived equation with the given options. We have also answered some additional questions about circle equations.