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Chapter 10: Problem 45

Write the equation for each circle described. Center \((0,0)\) and passing through \((4,5)\)

Short Answer

Expert verified
The equation of the circle is \((x^2 + y^2 = 41)\).

Step by step solution

01

Recall the Standard Equation of a Circle

The standard equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Since the center is given as \( (0, 0) \), the equation simplifies to \((x^2 + y^2 = r^2)\).
02

Calculate the Radius

To find the radius \((r)\), use the distance formula between the center \( (0,0) \) and the point \( (4,5) \). The radius can be calculated as: \[r = \sqrt{(4 - 0)^2 + (5 - 0)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]
03

Write the Final Equation

Now that the radius \((r)\) is known, substitute \((r = \sqrt{41})\) back into the simplified circle equation, giving: \((x^2 + y^2 = 41)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Equation of a Circle
The standard equation of a circle is a fundamental concept in geometry. This equation represents all the points \((x, y)\) that lie on the circle, given a specific center \((h, k)\) and radius \(r\). It is expressed as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
  • \((h, k)\) is the center of the circle.
  • \(r\) is the radius of the circle.

For example, if the center of a circle is at the origin, \((0,0)\), the equation simplifies to:
\[ x^2 + y^2 = r^2 \]
This form shows that every point \((x, y)\) on the circle is at a distance \(r\) from the center (0,0).
Distance Formula
The distance formula is used to find the distance between two points in the coordinate system. It is derived from the Pythagorean theorem and can be stated as:
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
  • \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

In the exercise example, we used the distance formula to calculate the radius \(r\) of the circle. The center is \((0,0)\) and the given point is \((4,5)\). Plug in the values:
\[\text{Radius} = \sqrt{(4 - 0)^2 + (5 - 0)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]
Therefore, the radius of the circle is \(\text{sqrt}(41)\).
Radius Calculation
The radius is an essential part of a circle's equation. Knowing how to calculate it accurately is key. As demonstrated, the radius is derived using the distance formula between the center of the circle and any point on the circle. In practice:
1. Identify the center \((h, k)\) and a point on the circle \((x, y)\).
2. Apply the distance formula to these points.

Using our exercise example, for center \((0,0)\) and point \((4,5)\):
\[\text{Radius} = \sqrt{(4 - 0)^2 + (5 - 0)^2} = \sqrt{41} \]
This value, \(\text{sqrt}(41)\), is then used in the simplified circle equation:
\[ x^2 + y^2 = 41 \]
This equation now represents the circle with the given center and radius.

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Most popular questions from this chapter

Use a graphing calculator to solve each problem. Graph \(y=x^{2}\) using the viewing window with \(-1 \leq x \leq 1\) and \(0 \leq y \leq 1\). Graph \(y=2 x^{2}-4 x+5\) using the viewing window with \(-1 \leq x \leq 3\) and \(3 \leq y \leq 11 .\) What can you say about the two graphs?

Find all points on the ellipse \(9 x^{2}+25 y^{2}=225\) that are twice as far from one focus as they are from the other focus.

Write each of the following equations in one of the forms: \(y=a(x-h)^{2}+k, \quad x=a(y-h)^{2}+k\) \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1,\) or \((x-h)^{2}+(y-k)^{2}=r^{2}\). Then identify each equation as the equation of a parabola, an ellipse, or a circle. $$2 x^{2}+4 y^{2}=4-y$$

Write each of the following equations in one of the forms: \(y=a(x-h)^{2}+k, \quad x=a(y-h)^{2}+k\) \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1,\) or \((x-h)^{2}+(y-k)^{2}=r^{2}\). Then identify each equation as the equation of a parabola, an ellipse, or a circle. $$4 x^{2}+12 y^{2}=4$$

Find the center and radius of each circle. $$x^{2}+y^{2}+2 y=8$$
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