Conic Sections: THE ELLIPSE
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1 Conic Sections: THE ELLIPSE An ellipse is the set of all points,such that the sum of the distance between, and two distinct points is a constant. These two distinct points are called the foci (plural of focus) of the ellipse and lie on the focal axis. The point on the focal axis midway between the foci is the centre. The points where the ellipse intersects its axis are the vertices of the ellipse. Standard Form of an Equation of an Ellipse Centered at the Origin An ellipse with the centre at the origin with a major axis along the axis and a minor axis along the axis has the equation: Label the FOCI, CENTRE, VERTEX, AND FOCAL AXIS (major axis) where and are positive real numbers. In standard form of the equation, the right side must equal to. A. Graphing an Ellipse To graph an ellipse centered at the origin, find the and intercepts. To find the intercepts, let 0. To find the intercepts, let 0. The intercepts are: The intercepts are: Standard Form of an Equation of an Ellipse Centered at the Origin An ellipse with the centre at the origin with a major axis along the axis and a minor axis along the axis has the equation: where and are positive real numbers. In standard form of the equation, the right side must equal to.
2 Example. Graph the ellipse given by the equation: a. b d A circle is a special case of an ellipse where! ". Therefore, it is not surprising that we graph an ellipse centered at the point #,$ in much the same way we graph a circle. B. Graphing an Ellipse Whose Centre is NOT at the Origin Standard Form of an Equation of an Ellipse Centered at Any Point An ellipse with the centre at any point %,& with a major axis of length '! parallel to the axis and a minor axis of length 2 parallel to the axis has the equation: () (* where and are positive real numbers. In standard form of the equation, the right side must equal to.
3 Example 2. Graph the ellipse given by the equation: a. ( +, b. + ( c C. The Geometry of an Ellipse Both figures show a point -, of an ellipse. The fixed points. and. are the foci of the ellipse, and the distances whose sum is constant are / and /. Structure of an Ellipse: Constructing an Ellipse: pencil, loop of string, 2 pins Put the loop around the two pins placed at. and., pull the string taut with a pencil at point -, and move the pencil around to trace out the ellipse. Question: The sum of 0 and 0 ' is constant BUT what is it equal to? / + / = constant ANS: ' '! Deriving the Standard Form of the Equation for an Ellipse: Using the distance formula, the equation becomes: Rearrange by eliminating radicals: AXIS Alert: For an ellipse, the word axis is used in several ways. The FOCAL AXIS is a line. The MAJOR and MINOR axes are line segments. The SEMIMAJOR and SEMIMINOR AXES are numbers.
4 As with circles, a line segment with endpoints on an ellipse is a CHORD of the ellipse. The chord lying on the focal axis is the MAJOR AXIS of the ellipse. The chord through the centre perpendicular to the focal axis is the MINOR AXIS. The length of the major axis is 2, and the minor axis is 2. The number is the SEMIMAJOR AXIS, and is the SEMIMINOR AXIS. D. Finding and Using the Vertices and Foci from an Elliptical Equation Example 3. Find the vertices and the foci of the ellipse: Example 4. Find an equation of the ellipse with foci 0,3 and 0,3 whose minor axis has length 4. E. Translations of Ellipses Translations do not change the length of the major or minor axis or the Pythagorean relation.
5 Example 5. Find the standard form of the equation for the ellipse whose major axis has endpoints 2, and 8,, and whose minor axis has a length of 8. Example 6. Find the centre, vertices and foci of the ellipse: + (8 Extra Practice:. Write the equation for the ellipse in standard form and general form. 2. Write the equation in standard form and general form for an ellipse with a centre at 2,5 and passing through 5,5,,5,2,2, and 2,2
6 3. Find the coordinates of the centre, the length of the major and minor axes, and the coordinates of the foci of the following ellipse: Find the coordinates of the centre, the length of the major and minor axes, and the coordinates of the foci of the following ellipse:
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