%5E2%2B(y%2B5)%5E2%3D16.jpeg "(x-7)^2+(y+5)^2=16")
Understanding the Equation: (x - 7)^2 + (y + 5)^2 = 16
The equation (x - 7)^2 + (y + 5)^2 = 16 represents a circle in the coordinate plane. Let's break down why:
The Standard Equation of a Circle
The general form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
Analyzing our Equation
Comparing our equation, (x - 7)^2 + (y + 5)^2 = 16, to the standard form, we can identify the following:
- Center: (h, k) = (7, -5)
- Radius: r^2 = 16, so r = 4
Visualizing the Circle
Now that we know the center and radius, we can visualize the circle on a graph:
- Plot the center: Locate the point (7, -5) on the coordinate plane.
- Draw the circle: With the center as the starting point, use the radius of 4 units to draw a circle. You can mark points 4 units away from the center in all directions (up, down, left, right) and connect them to form the circle.
Properties and Applications
This equation tells us that every point (x, y) on the circle is exactly 4 units away from the point (7, -5). This property has applications in various fields:
- Geometry: Circles are fundamental shapes in geometry, used in constructions and calculations.
- Physics: Circles appear in concepts like circular motion and orbits.
- Engineering: Circular shapes are used in various structures and machines.
Understanding the equation of a circle allows us to analyze and work with these shapes in different contexts.