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To determine the equation of the circle graphed, we need to identify the center and the radius of the circle. 1. **Identify the center of the circle:** From the graph, the center of the circle is at the origin \((0, 0)\). 2. **Determine the radius of the circle:** The radius is the distance from the center to any point on the circle. From the graph, the circle intersects the x-axis at \(x = 5\) and \(x = -5\), and the y-axis at \(y = 5\) and \(y = -5\). Therefore, the radius \(r\) is 5 units. The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is: \[ (x - h)^2 + (y - k)^2 = r^2\] Since the center is \((0, 0)\) and the radius is 5, the equation of the circle is: \[ (x - 0)^2 + (y - 0)^2 = 5^2\] Simplifying, we get: \[ x^2 + y^2 = 25\] So, the equation of the circle is: \[ x^2 + y^2 = 25\]

Question

To determine the equation of the circle graphed, we need to identify the center and the radius of the circle. 1. Identify the center of the circle: From the graph, the center of the circle is at the origin (0,0)(0, 0). 2. Determine the radius of the circle: The radius is the distance from the center to any point on the circle. From the graph, the circle intersects the x-axis at x=5x = 5 and x=−5x = -5, and the y-axis at y=5y = 5 and y=−5y = -5. Therefore, the radius rr is 5 units. The standard form of the equation of a circle with center (h,k)(h, k) and radius rr is: (x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2 Since the center is (0,0)(0, 0) and the radius is 5, the equation of the circle is: (x−0)2+(y−0)2=52 (x - 0)^2 + (y - 0)^2 = 5^2 Simplifying, we get: x2+y2=25 x^2 + y^2 = 25 So, the equation of the circle is: x2+y2=25 x^2 + y^2 = 25

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Solution

Para determinar la ecuación del círculo graficado, necesitamos identificar el centro y el radio del círculo.

  1. Identificar el centro del círculo: Desde el gráfico, el centro del círculo está en el origen (0,0)(0, 0).

  2. Determinar el radio del círculo: El radio es la distancia desde el centro hasta cualquier punto en el círculo. Desde el gráfico, el círculo intersecta el eje x en x=5x = 5 y x=−5x = -5, y el eje y en y=5y = 5 y y=−5y = -5. Por lo tanto, el radio rr es de 5 unidades.

La forma estándar de la ecuación de un círculo con centro (h,k)(h, k) y radio rr es:

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

Dado que el centro es (0,0)(0, 0) y el radio es 5, la ecuación del círculo es:

(x−0)2+(y−0)2=52 (x - 0)^2 + (y - 0)^2 = 5^2

Simplificando, obtenemos:

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

Por lo tanto, la ecuación del círculo es:

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

This problem has been solved

Similar Questions

To determine which equation represents the circle shown in the figure, we need to identify the center and the radius of the circle. 1. **Identify the center of the circle:** From the graph, the center of the circle is at \((-1, 1)\). 2. **Identify the radius of the circle:** The radius can be determined by measuring the distance from the center to any point on the circle. From the graph, the radius is 2 units. 3. **Write the standard form of the equation of a circle:** The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] 4. **Substitute the center and radius into the equation:** The center is \((-1, 1)\) and the radius is 2. Therefore: \[ (x - (-1))^2 + (y - 1)^2 = 2^2 \] Simplifying, we get: \[ (x + 1)^2 + (y - 1)^2 = 4 \] 5. **Match with the given options:** The correct equation is: \[ (x + 1)^2 + (y - 1)^2 = 4 \] Therefore, the correct answer is: \[ \boxed{D} \]

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