Math Problem Statement

The diagram shows a circle with centre C. The circle intersects the axes at (0, -2), (0, -10) and (-10, 0) as shown on the diagram. Work out the equation of the circle.

Solution

To find the equation of the circle, let's first gather the necessary information from the diagram.

Step 1: The general equation of a circle

The equation of a circle with center (h,k)(h, k) and radius rr is given by:

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

Where:

  • (h,k)(h, k) is the center of the circle.
  • rr is the radius.

Step 2: Determining the center

From the diagram, it appears the circle is centered at (5,5)(-5, -5). This is determined by the points where the circle intersects the axes. The x- and y-intercepts of the circle are given as (0,2)(0, -2), (0,10)(0, -10), (10,0)(-10, 0), and (0,10)(0, -10). The point at (5,5)(-5, -5) looks like the center.

Step 3: Determining the radius

The radius is the distance from the center to any point on the circle. Let's use the point (0,10)(0, -10), which is on the circle, and calculate the distance to the center (5,5)(-5, -5):

Radius=(x2x1)2+(y2y1)2\text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Substitute the coordinates (x1,y1)=(5,5)(x_1, y_1) = (-5, -5) and (x2,y2)=(0,10)(x_2, y_2) = (0, -10):

Radius=(0(5))2+(10(5))2=52+(5)2=25+25=50=52\text{Radius} = \sqrt{(0 - (-5))^2 + (-10 - (-5))^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}

Step 4: Equation of the circle

Now that we know the center is (5,5)(-5, -5) and the radius is 525\sqrt{2}, we can write the equation of the circle:

(x+5)2+(y+5)2=(52)2(x + 5)^2 + (y + 5)^2 = (5\sqrt{2})^2

Simplifying:

(x+5)2+(y+5)2=50(x + 5)^2 + (y + 5)^2 = 50

Thus, the equation of the circle is:

(x+5)2+(y+5)2=50(x + 5)^2 + (y + 5)^2 = 50

Would you like further clarification or details on any part of the solution? Here are some related questions for deeper understanding:

  1. How do you calculate the distance between two points on the coordinate plane?
  2. How would you find the radius if only one intercept point and the center are given?
  3. What if the center of the circle was located on the origin? How would the equation change?
  4. How can you apply the distance formula in other geometric contexts, such as in triangles or polygons?
  5. What is the significance of the center and radius in relation to the graph of the circle?

Tip: When solving for the equation of a circle, always double-check that you have correctly identified the center and radius, as these are key to writing the equation.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Equation of a Circle
Distance Formula

Formulas

Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2)

Theorems

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Suitable Grade Level

Grades 9-12