The Circle Below Is Centered At The Point \[$(4, -3)\$\] And Has A Radius Of Length 3. What Is Its Equation?A. \[$(x-3)^2+(y+4)^2=9\$\]B. \[$(x+4)^2+(y-3)^2=3^2\$\]C. \[$(x-4)^2+(y+3)^2=3^2\$\]D.

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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 article, we will explore how to find the equation of a circle given its center and radius.

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 r{r} is the radius.

Finding the Equation of a Circle

Given that the circle is centered at the point (4,−3){(4, -3)} and has a radius of length 3, we can plug these values into the general equation of a circle.

Step 1: Identify the Center and Radius

The center of the circle is given as (4,−3){(4, -3)}, which means that h=4{h = 4} and k=−3{k = -3}. The radius of the circle is given as 3, which means that r=3{r = 3}.

Step 2: Plug in the Values

Now that we have identified the center and radius, we can plug these values into the general equation of a circle.

(x−4)2+(y−(−3))2=32{(x-4)^2 + (y-(-3))^2 = 3^2}

Step 3: Simplify the Equation

Simplifying the equation, we get:

(x−4)2+(y+3)2=9{(x-4)^2 + (y+3)^2 = 9}

Conclusion

The equation of the circle is (x−4)2+(y+3)2=9{(x-4)^2 + (y+3)^2 = 9}. This equation represents a circle with a center at (4,−3){(4, -3)} and a radius of length 3.

Comparing with the Options

Let's compare our equation with the options given:

A. (x−3)2+(y+4)2=9{(x-3)^2+(y+4)^2=9} B. (x+4)2+(y−3)2=32{(x+4)^2+(y-3)^2=3^2} C. (x−4)2+(y+3)2=32{(x-4)^2+(y+3)^2=3^2} D. (x+3)2+(y−4)2=9{(x+3)^2+(y-4)^2=9}

Our equation matches option C.

The Final Answer

The final answer is option C: (x−4)2+(y+3)2=32{(x-4)^2+(y+3)^2=3^2}.

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 article, we will explore some frequently asked questions about the circle equation.

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 r{r} is the radius.

Q: How do I find the equation of a circle given its center and radius?

A: To find the equation of a circle given its center and radius, you can plug the values into the general equation of a circle. For example, if the center is (4,−3){(4, -3)} and the radius is 3, the equation would be:

(x−4)2+(y−(−3))2=32{(x-4)^2 + (y-(-3))^2 = 3^2}

Q: What is the significance of the center and radius in the circle equation?

A: The center and radius are the two most important components of the circle equation. The center represents the point around which the circle is centered, while the radius represents the distance from the center to any point on the circle.

Q: How do I simplify the circle equation?

A: To simplify the circle equation, you can start by expanding the squared terms and then combining like terms. For example, if the equation is:

(x−4)2+(y+3)2=9{(x-4)^2 + (y+3)^2 = 9}

You can simplify it by expanding the squared terms and combining like terms:

x2−8x+16+y2+6y+9=9{x^2 - 8x + 16 + y^2 + 6y + 9 = 9}

Q: What is the difference between the circle equation and the standard form of a circle?

A: The circle equation is the general form of a circle, while the standard form of a circle is a specific type of circle equation that is written in the form:

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

The standard form is a more concise way of writing the circle equation, but it is still the same equation.

Q: Can I use the circle equation to find the center and radius of a circle?

A: Yes, you can use the circle equation to find the center and radius of a circle. By rearranging the equation, you can isolate the center and radius terms. For example, if the equation is:

(x−4)2+(y+3)2=9{(x-4)^2 + (y+3)^2 = 9}

You can rearrange it to find the center and radius:

h=4{h = 4} k=−3{k = -3} r=9=3{r = \sqrt{9} = 3}

Conclusion

The circle equation is a powerful tool for working with circles. By understanding the basics of the circle equation, you can solve a wide range of problems involving circles. Whether you are working with geometry, trigonometry, or calculus, the circle equation is an essential tool to have in your toolkit.

Frequently Asked Questions

  • 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}.
  • Q: How do I find the equation of a circle given its center and radius? A: To find the equation of a circle given its center and radius, you can plug the values into the general equation of a circle.
  • Q: What is the significance of the center and radius in the circle equation? A: The center and radius are the two most important components of the circle equation.
  • Q: How do I simplify the circle equation? A: To simplify the circle equation, you can start by expanding the squared terms and then combining like terms.
  • Q: What is the difference between the circle equation and the standard form of a circle? A: The circle equation is the general form of a circle, while the standard form of a circle is a specific type of circle equation that is written in the form (x−h)2+(y−k)2=r2{(x-h)^2 + (y-k)^2 = r^2}.
  • Q: Can I use the circle equation to find the center and radius of a circle? A: Yes, you can use the circle equation to find the center and radius of a circle.