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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: \[ \boxed{D} \]

Question

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)(-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)(h, k) and radius rr is: (x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2 4. Substitute the center and radius into the equation: The center is (−1,1)(-1, 1) and the radius is 2. Therefore: (x−(−1))2+(y−1)2=22 (x - (-1))^2 + (y - 1)^2 = 2^2 Simplifying, we get: (x+1)2+(y−1)2=4 (x + 1)^2 + (y - 1)^2 = 4 5. Match with the given options: The correct equation is: (x+1)2+(y−1)2=4 (x + 1)^2 + (y - 1)^2 = 4 Therefore, the correct answer is: D \boxed{D}

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Solution

Para determinar qué ecuación representa el círculo mostrado en la figura, necesitamos identificar el centro y el radio del círculo.

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

  2. Identificar el radio del círculo: El radio se puede determinar midiendo la distancia desde el centro hasta cualquier punto en el círculo. Desde el gráfico, el radio es de 2 unidades.

  3. Escribir la forma estándar de la ecuación de un círculo: La forma estándar de la ecuación de un círculo con centro (h, k) y radio r es: (x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2

  4. Sustituir el centro y el radio en la ecuación: El centro es (-1, 1) y el radio es 2. Por lo tanto: (x−(−1))2+(y−1)2=22 (x - (-1))^2 + (y - 1)^2 = 2^2 Simplificando, obtenemos: (x+1)2+(y−1)2=4 (x + 1)^2 + (y - 1)^2 = 4

  5. Coincidir con las opciones dadas: La ecuación correcta es: (x+1)2+(y−1)2=4 (x + 1)^2 + (y - 1)^2 = 4 Por lo tanto, la respuesta correcta es: D \boxed{D}

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Similar Questions

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: \[ x^2 + y^2 = 25\]

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The diagram shows a circle graphed in the x y coordinate plane that has center C at coordinates 1 comma negative 2 and passes through point P at coordinates 6 comma negative 7.A circle with its center at   passes through point   and is graphed in the  -plane. Which of the following is an equation for the graph of the circle?

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