If A Circle Has An Area Of \[$110.25\pi\$\] And Its Center Is At The Origin, Determine The Equation For This Circle And Find 3 Points That Are On The Circle.

If a Circle Has an Area of ${110.25\pi\$} and Its Center is at the Origin, Determine the Equation for This Circle and Find 3 Points That Are on the Circle

In this article, we will explore the concept of circles and their equations. We will start by understanding the relationship between the area of a circle and its radius. Then, we will use this relationship to determine the equation of a circle with a given area. Finally, we will find three points that lie on this circle.

The area of a circle is given by the formula:

$$A = \pi r^2$$

where $A$ is the area of the circle and $r$ is the radius of the circle. This formula shows that the area of a circle is directly proportional to the square of its radius.

Given that the area of the circle is ${110.25\pi\$}, we can use the formula above to determine the radius of the circle:

$$110.25\pi = \pi r^2$$

Dividing both sides of the equation by $\pi$, we get:

$$110.25 = r^2$$

Taking the square root of both sides of the equation, we get:

$$r = \sqrt{110.25}$$

$$r = 10.5$$

Now that we have the radius of the circle, we can determine the equation of the circle. The equation of a circle with its center at the origin is given by:

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

Substituting the value of $r$ that we found above, we get:

$$x^2 + y^2 = 10.5^2$$

$$x^2 + y^2 = 110.25$$

This is the equation of the circle.

To find three points that lie on the circle, we can use the equation of the circle and substitute different values of $x$ and $y$. Let's find three points that lie on the circle:

• Point 1: $(x, y) = (0, 10.5)$

  Substituting $x = 0$ and $y = 10.5$ into the equation of the circle, we get:

  $$0^2 + 10.5^2 = 110.25$$

  This is true, so the point $(0, 10.5)$ lies on the circle.

• Point 2: $(x, y) = (10.5, 0)$

  Substituting $x = 10.5$ and $y = 0$ into the equation of the circle, we get:

  $$10.5^2 + 0^2 = 110.25$$

  This is true, so the point $(10.5, 0)$ lies on the circle.

• Point 3: $(x, y) = (5, 8.5)$

  Substituting $x = 5$ and $y = 8.5$ into the equation of the circle, we get:

  $$5^2 + 8.5^2 = 25 + 72.25 = 97.25$$

  This is not equal to $110.25$, so the point $(5, 8.5)$ does not lie on the circle.

However, we can find a point that lies on the circle by using the equation of the circle and the distance formula. The distance formula is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

We can use this formula to find the distance between the point $(5, 8.5)$ and the center of the circle $(0, 0)$:

$$d = \sqrt{(5 - 0)^2 + (8.5 - 0)^2}$$

$$d = \sqrt{25 + 72.25}$$

$$d = \sqrt{97.25}$$

$$d = 9.85$$

The distance between the point $(5, 8.5)$ and the center of the circle $(0, 0)$ is $9.85$. We can use this distance to find the point on the circle that is closest to the point $(5, 8.5)$. Let's call this point $(x, y)$. We can use the equation of the circle to find the value of $x$ and $y$:

$$x^2 + y^2 = 110.25$$

$$x^2 + (9.85 - 5)^2 = 110.25$$

$$x^2 + 4.85^2 = 110.25$$

$$x^2 + 23.4025 = 110.25$$

$$x^2 = 86.8475$$

$$x = \sqrt{86.8475}$$

$$x = 9.32$$

Now that we have the value of $x$, we can find the value of $y$:

$$y^2 = 110.25 - x^2$$

$$y^2 = 110.25 - 86.8475$$

$$y^2 = 23.4025$$

$$y = \sqrt{23.4025}$$

$$y = 4.85$$

So, the point $(x, y) = (9.32, 4.85)$ lies on the circle.

In this article, we determined the equation of a circle with a given area and found three points that lie on the circle. We used the formula for the area of a circle to determine the radius of the circle, and then used the equation of a circle to find the equation of the circle. We also used the distance formula to find the point on the circle that is closest to a given point.
Q&A: If a Circle Has an Area of ${110.25\pi\$} and Its Center is at the Origin, Determine the Equation for This Circle and Find 3 Points That Are on the Circle

A: The area of a circle is given by the formula:

$$A = \pi r^2$$

where $A$ is the area of the circle and $r$ is the radius of the circle. This formula shows that the area of a circle is directly proportional to the square of its radius.

A: To determine the equation of a circle, you need to know the radius of the circle. The equation of a circle with its center at the origin is given by:

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

where $r$ is the radius of the circle.

A: To find the radius of a circle given its area, you can use the formula:

$$r = \sqrt{\frac{A}{\pi}}$$

where $A$ is the area of the circle.

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A: To find the equation of the circle, we need to determine the radius of the circle. We can use the formula:

$$r = \sqrt{\frac{A}{\pi}}$$

Substituting the value of $A$ that we are given, we get:

$$r = \sqrt{\frac{110.25\pi}{\pi}}$$

$$r = \sqrt{110.25}$$

$$r = 10.5$$

Now that we have the radius of the circle, we can determine the equation of the circle:

$$x^2 + y^2 = 10.5^2$$

$$x^2 + y^2 = 110.25$$

This is the equation of the circle.

A: To find three points that lie on the circle, we can use the equation of the circle and substitute different values of $x$ and $y$. Let's find three points that lie on the circle:

• Point 1: $(x, y) = (0, 10.5)$

  Substituting $x = 0$ and $y = 10.5$ into the equation of the circle, we get:

  $$0^2 + 10.5^2 = 110.25$$

  This is true, so the point $(0, 10.5)$ lies on the circle.

• Point 2: $(x, y) = (10.5, 0)$

  Substituting $x = 10.5$ and $y = 0$ into the equation of the circle, we get:

  $$10.5^2 + 0^2 = 110.25$$

  This is true, so the point $(10.5, 0)$ lies on the circle.

• Point 3: $(x, y) = (5, 8.5)$

  Substituting $x = 5$ and $y = 8.5$ into the equation of the circle, we get:

  $$5^2 + 8.5^2 = 25 + 72.25 = 97.25$$

  This is not equal to $110.25$, so the point $(5, 8.5)$ does not lie on the circle.

However, we can find a point that lies on the circle by using the equation of the circle and the distance formula. The distance formula is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

We can use this formula to find the distance between the point $(5, 8.5)$ and the center of the circle $(0, 0)$:

$$d = \sqrt{(5 - 0)^2 + (8.5 - 0)^2}$$

$$d = \sqrt{25 + 72.25}$$

$$d = \sqrt{97.25}$$

$$d = 9.85$$

The distance between the point $(5, 8.5)$ and the center of the circle $(0, 0)$ is $9.85$. We can use this distance to find the point on the circle that is closest to the point $(5, 8.5)$. Let's call this point $(x, y)$. We can use the equation of the circle to find the value of $x$ and $y$:

$$x^2 + y^2 = 110.25$$

$$x^2 + (9.85 - 5)^2 = 110.25$$

$$x^2 + 4.85^2 = 110.25$$

$$x^2 + 23.4025 = 110.25$$

$$x^2 = 86.8475$$

$$x = \sqrt{86.8475}$$

$$x = 9.32$$

Now that we have the value of $x$, we can find the value of $y$:

$$y^2 = 110.25 - x^2$$

$$y^2 = 110.25 - 86.8475$$

$$y^2 = 23.4025$$

$$y = \sqrt{23.4025}$$

$$y = 4.85$$

So, the point $(x, y) = (9.32, 4.85)$ lies on the circle.

A: The center of the circle being at the origin means that the equation of the circle is given by:

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

This is a simpler equation than the equation of a circle with its center at a different point.

A: To find the distance between two points on a circle, you can use the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

This formula gives the distance between the two points.

A: The diameter of a circle is twice the radius of the circle. The formula for the diameter of a circle is:

$$d = 2r$$

where $d$ is the diameter of the circle and $r$ is the radius of the circle.