Equation of a Circle
Posted by Professor Cram in Conic Sections
Conic Sections: An Introduction
A conic section is a curve formed by the intersection of a cone with a plane. Depending on how the plane is oriented, the curve will be one of the conic sections — circle, ellipse, parabola, or hyperbola:
- A circle is the set of all points that area equally distant from a fixed point C, the center of the circle.
- An ellipse is the set of all point surrounding two foci, or focus points, such that the sum of the distances from any point to each focus remains constant. An ellipse can be oriented vertically (shaped higher than wide) or horizontally (shaped wider than high).
- A parabola is the set of points that are equqally distant from the focus point and the directrix, a fixed line. A parabola can be oriented vertically (opening up or down) or horizontally (opening left or right).
- A hyperbola is the set of all points around two foci, or focus points, such that the difference of the distances from any point to each focus remains constant. A hyperbola can be oriented vertically (opening up and down) or horizontally (opening left and right).
Equation of a Circle
This Tab Tutor program will show you how to find the equation of a circle from its center and radius, and the radius and center of a circle from its equation.
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can somebody help me with this problem?????
Find the center of the circle with these three points: (1,10),(2,11), and (5,12) Seperate the coordinates with a comma
A conic section is not a perfect ellipse which is a cylindric section.
Liz, let the center be (a, b)
Use the distance formula d=√((x_2-x_1 )^2+(y_2-y_1 )^2 )
to find the distance between (1, 10) and (a, b) and the distance between (2, 11) and (a, b). Since these are both equal to the radius of the circle, they can be put equal. This leads to the equation
a + b = 12 (1)
Do the same for the points (1, 10) and (5, 12) to obtain a second equation:
2a + b = 17 (2)
Solve eqations (1) and (2) simultaneously to get a = 5 and b = 7
Peter is incorrect. The ellipse (from a conic section or whatever) is perfect … else it would not be called an ellipse. It is symmetric about both axes, etc, etc.
But when seen in relation to the cone, the ellipse’s “center” (where axes intersect) does not match the cone’s central axis. This may be the root of his confusion. He may think the ellipse is centered on the cone and thinks “the upper part must be smaller as the radius gets smaller”.
He is, however, correct that a cylindric section is also an ellipse. In projective geometry, a cylinder is considered to be a cone whose apex is at infinity.
A conic section is symmetrical North-South but not East-West and for that reason cannot be a perfect ellipse. The origin of the word ellipse goes back to Apollonius of Perga who used it for his purposes to describe a conic section not a cylindric section, that is why I use the expression perfect ellipse for a cylindric section.