Circles and ellipses are examples of conic sections, which are curves formed by the intersection of a plane with a cone. The flower bed is 15 feet wide, and 15 feet long. You are using a circular sprinkler system, and the water reaches 6 feet out from the center. The sprinkler is located, from the bottom left corner of the bed, 7 feet up, and 6 feet over. The water reaches 6 feet out from the sprinkler, so the circle radius is 6 feet.
Therefore the equation of this circle is:. The first step to finding the percentage of the garden that is being watered is to check that none of the water is falling outside the garden.
The center of the circle can be found by comparing the equation in this exercise to the equation of a circle:. There are many points you could choose. Plugging this into the equation, we get:. The left side is equal to the right side of the equation, and so this is a valid point on the circle. Now, complete the square in both parentheses, subtracting or adding the necessary constant to both sides of the equation:.
It even has a number on the right side. How can we get rid of them to get into standard form? Privacy Policy. Skip to main content. Conic Sections. Search for:. The Circle and the Ellipse. Learning Objectives Explain how the equation of a circle describes its properties. Key Takeaways Key Points A circle is defined as the set of points that are a fixed distance from a center point. The distance formula can be extended directly to the definition of a circle by noting that the radius is the distance between the center of a circle and the edge.
Key Terms diameter : Two times the radius of a circle. The midpoint of the major axis is called the center of the ellipse. A line perpendicular to the major axis and passes through the center is called the minor axis of the ellipse. These two are the diameters of the ellipse. The major axis is the longer diameter, and the minor axis is the shorter diameter. One-half of the major and the minor axis are known as the semi-major axis and the semi-minor axis, respectively. The circle is the locus of a point, which moves with an equidistance from a given fixed point.
The distance between any point on the circle and its center is constant, which is known as the radius. A circle is formed when a plane intersect a cone, perpendicular to its axis. Mathematically, a circle is a major shape in the field of geometry and its definition states: a circle is a shape with all points at the same distance from its center.
It is named by its center. Some real world examples of a circle are wheels, a dinner plate and the surface of a coin. The circle has been known before recorded history. The Sun and the Moon are natural examples of a circle, whereas even a short stalk blowing in the wind, forms the shape of a circle in the sand. The principle of a circle was applied in the formation of wheels and gears by the pre-historic man. Now in the modern era, there are enumerable varieties of machineries based on the circle shape.
The study of a circle and its development is applicable in the fields of mathematics, geometry, astronomy and calculus. Radius : a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter. Diameter : a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle.
It is a special case of a chord, namely the longest chord, and it is twice the radius. Semicircl e: a region bounded by a diameter and an arc lying between the diameter's endpoints. It is a special case of a circular segment, namely the largest one.
Circular sector : a region bounded by two radii and an arc lying between the radii. Mathematically, an ellipse is a common shape in the field of mathematics.
Its definition states: a curved line forming a closed loop, where the sum of the distances from two points foci to every point on the line is constant. The real life examples of an ellipse are: a hula hoop, a glass of water, and a simple dinner plate when tilted to view at an angle.
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