Which Equation Represents A Circle With A Center At ( − 3 , − 5 (-3, -5 ( − 3 , − 5 ] And A Radius Of 6 Units?A. ( X − 3 ) 2 + ( Y − 5 ) 2 = 6 (x - 3)^2 + (y - 5)^2 = 6 ( X − 3 ) 2 + ( Y − 5 ) 2 = 6 B. ( X − 3 ) 2 + ( Y − 5 ) 2 = 36 (x - 3)^2 + (y - 5)^2 = 36 ( X − 3 ) 2 + ( Y − 5 ) 2 = 36 C. ( X + 3 ) 2 + ( Y + 5 ) 2 = 6 (x + 3)^2 + (y + 5)^2 = 6 ( X + 3 ) 2 + ( Y + 5 ) 2 = 6 D. ( X + 3 ) 2 + ( Y + 5 ) 2 = 36 (x + 3)^2 + (y + 5)^2 = 36 ( X + 3 ) 2 + ( Y + 5 ) 2 = 36

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The equation of a circle is a fundamental concept in mathematics, and it is used to represent a set of points that are equidistant from a central point called the center. In this article, we will explore the equation of a circle and determine which of the given options represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units.

The General Equation of a Circle

The general equation of a circle is given by:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

where (h,k)(h, k) is the center of the circle and rr is the radius.

Understanding the Given Options

We are given four options, and we need to determine which one represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units.

  • Option A: (x−3)2+(y−5)2=6(x - 3)^2 + (y - 5)^2 = 6
  • Option B: (x−3)2+(y−5)2=36(x - 3)^2 + (y - 5)^2 = 36
  • Option C: (x+3)2+(y+5)2=6(x + 3)^2 + (y + 5)^2 = 6
  • Option D: (x+3)2+(y+5)2=36(x + 3)^2 + (y + 5)^2 = 36

Analyzing the Options

Let's analyze each option and determine which one represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units.

Option A: (x−3)2+(y−5)2=6(x - 3)^2 + (y - 5)^2 = 6

This option represents a circle with a center at (3,5)(3, 5), not (−3,−5)(-3, -5). Therefore, this option is not correct.

Option B: (x−3)2+(y−5)2=36(x - 3)^2 + (y - 5)^2 = 36

This option represents a circle with a center at (3,5)(3, 5) and a radius of 6 units. However, the center of the circle is not at (−3,−5)(-3, -5), so this option is not correct.

Option C: (x+3)2+(y+5)2=6(x + 3)^2 + (y + 5)^2 = 6

This option represents a circle with a center at (−3,−5)(-3, -5), but the radius is not 6 units. The radius is actually 6\sqrt{6} units, which is not equal to 6 units. Therefore, this option is not correct.

Option D: (x+3)2+(y+5)2=36(x + 3)^2 + (y + 5)^2 = 36

This option represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units. The equation is in the correct form, and the center and radius match the given conditions. Therefore, this option is correct.

Conclusion

In conclusion, the equation that represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units is:

(x+3)2+(y+5)2=36(x + 3)^2 + (y + 5)^2 = 36

This equation is in the correct form, and the center and radius match the given conditions. Therefore, option D is the correct answer.

Key Takeaways

  • The equation of a circle is given by (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center of the circle and rr is the radius.
  • To determine which option represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units, we need to analyze each option and determine which one matches the given conditions.
  • The correct option is the one that represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units.

Final Answer

The final answer is:

In the previous article, we discussed the equation of a circle and determined which option represents a circle with a center at (−3,−5)(-3, -5) and a radius of 6 units. In this article, we will answer some frequently asked questions about the equation of a circle.

Q: What is the general equation of a circle?

A: The general equation of a circle is given by:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

where (h,k)(h, k) is the center of the circle and rr is the radius.

Q: How do I determine the center and radius of a circle from its equation?

A: To determine the center and radius of a circle from its equation, you need to look at the equation and identify the values of hh, kk, and rr. The center of the circle is given by (h,k)(h, k), and the radius is given by rr.

Q: What is the significance of the center of a circle?

A: The center of a circle is the point that is equidistant from all points on the circle. It is the point around which the circle is centered.

Q: What is the significance of the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is a measure of the size of the circle.

Q: How do I graph a circle from its equation?

A: To graph a circle from its equation, you need to plot the center of the circle and then draw a circle with the given radius.

Q: What is the equation of a circle with a center at (0,0)(0, 0) and a radius of 5 units?

A: The equation of a circle with a center at (0,0)(0, 0) and a radius of 5 units is:

x2+y2=25x^2 + y^2 = 25

Q: What is the equation of a circle with a center at (2,3)(2, 3) and a radius of 4 units?

A: The equation of a circle with a center at (2,3)(2, 3) and a radius of 4 units is:

(x−2)2+(y−3)2=16(x - 2)^2 + (y - 3)^2 = 16

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

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

Q: What is the equation of a circle with a center at (−1,2)(-1, 2) and a radius of 3 units?

A: The equation of a circle with a center at (−1,2)(-1, 2) and a radius of 3 units is:

(x+1)2+(y−2)2=9(x + 1)^2 + (y - 2)^2 = 9

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics, and it is used to represent a set of points that are equidistant from a central point called the center. We have discussed the general equation of a circle, how to determine the center and radius of a circle from its equation, and how to graph a circle from its equation. We have also answered some frequently asked questions about the equation of a circle.

Key Takeaways

  • The general equation of a circle is given by (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center of the circle and rr is the radius.
  • To determine the center and radius of a circle from its equation, you need to look at the equation and identify the values of hh, kk, and rr.
  • The center of a circle is the point that is equidistant from all points on the circle.
  • The radius of a circle is the distance from the center of the circle to any point on the circle.

Final Answer

The final answer is:

There is no final numerical answer to this problem. The article is a Q&A about the equation of a circle.