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?

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: \[ \boxed{D} \]

The equation below describes a circle. What are the coordinates of the center of the circle?(x - 6)2 + (y + 5)2 = 152A.(-6, -5)B.(-6, 5)C.(6, 5)D.(6, -5)SUBMITarrow_backPREVIOUS

The equation of a circle is given below. Identify the center and radius. Then graph the circle.=+x2−y224

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

2 A circle with centre C has equation .x y x y6 4 4 02 2+ - + + =(a) Find(i) the coordinates of C, [2](ii) the radius of the circle. [1](b) Determine the set of values of k for which the line y kx 3= - does not intersect or touch thecircle.