Problem 50 Determine the center and radius ... [FREE SOLUTION]

Chapter 1: Problem 50

Determine the center and radius of the circle with the given equation. $$x^{2}+y^{2}=49$$

Short Answer

Expert verified

The center of the circle is at the origin (0,0) and the radius is 7.

Step by step solution

Identify the center

In the given equation $ x^{2} + y^{2} = 49 $, there is no additional constant added or subtracted from x or y. Therefore, the center (h,k) of the circle is at the origin, (0,0).

Identify the radius

In the given equation $ x^{2} + y^{2} = 49 $, the constant on the right side is the square of the radius. Therefore, to find the radius r, take the square root of 49. The radius r is $ \sqrt{49} = 7 $.

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Key Concepts

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

Center of a Circle

The center of a circle is the point from which all points on the circle are equidistant. In mathematical terms, this is often denoted as $(h, k)$. For the equation of a circle $(x - h)^2 + (y - k)^2 = r^2$, $h$ and $k$ provide the coordinates of the center. When these values are both zero, such as in the equation $x^2 + y^2 = 49$, the circle is centered at the origin, $(0, 0)$.

To identify the center from a more complex equation, simply look for the values subtracted from $x$ and $y$. Those values tell you the horizontal and vertical shifts from the origin. If there is nothing subtracted, like in our example, the center remains at zero for both axes.

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its edge. In circle equations, the radius $r$ appears squared as $r^2$. To find the radius from the circle's equation $(x - h)^2 + (y - k)^2 = r^2$, you simply take the square root of the number on the equation's right side.

In our sample equation $x^2 + y^2 = 49$, just identify the number $49$ on the right. By taking the square root, you'll determine the radius: $\sqrt{49} = 7$. Thus, the radius of the circle described by this equation is 7. Remember, knowing the radius is essential as it indicates how large the circle is.

Standard Form of a Circle

The standard form of a circle's equation gives you crucial information about the circle, such as its center and radius. It is generally written as $(x - h)^2 + (y - k)^2 = r^2$. Here, $(h, k)$ is the center of the circle, and $r$ is its radius.

For our given equation $x^2 + y^2 = 49$, you will notice it fits this form with $h = 0$ and $k = 0$, and therefore $(h, k) = (0, 0)$. This shows the circle is centered at the origin. Also, because $r^2 = 49$, we find $r = 7$, making the radius 7. The standard form allows for straightforward identification of these traits, aiding in graphing and comprehending the circle's properties.

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91影视

Determine the center and radius of the circle with the given equation. $$x^{2}+y^{2}=49$$

Short Answer

Step by step solution

Identify the center

Identify the radius

Key Concepts

Center of a Circle

Radius of a Circle

Standard Form of a Circle

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