Major and Minor Axes
Major axis: The line segment through the foci of the ellipse with its endpoints on the ellipse, is called its major axis.
Minor Axis: The line segment through the centre and perpendicular to the major axis with its endpoints on the ellipse, is called its minor axis.
| Horizontal Ellipse | {x^2\over a^2}+{y^2\over b^2} = 1, (b < a) |
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| Vertex | (a, 0) and (-a, 0) |
| Centre | (0, 0) |
| Length of major axis | 2a |
| Length of major axis | 2b |
| Foci | (ae, 0) and (- ae, 0) |
| The equation of directories are | x = a\over e and x = – a\over e |
| Length of the latus rectum | 2b^2\over a |
| Eccentricity (e) | \sqrt {(1-{b^2\over a^2})} |
| Distance between foci | 2ae |
| Distance between directrices | 2a\over e |
| Vertical Ellipse | {x^2\over a^2}+{y^2\over b^2} = 1, (a < b) |
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| Vertex | (0, b) and (0, -b) |
| Centre | (0, 0) |
| Length of major axis | 2b |
| Length of major axis | 2a |
| Foci | (0, ae) and (0, -ae) |
| The equation of directories are | y = b\over e and x = – b\over e |
| Length of the latus rectum | 2a^2\over b |
| Eccentricity (e) | \sqrt {(1-{a^2\over b^2})} |
| Distance between foci | 2be |
| Distance between directrices | 2b\over e |
Position of a point concerning an ellipse
The point (x1, y1) lies outside, on or inside the ellipse
| {x_1^2\over a^2}+{y_1^2\over b^2} – 1 > 0 | outside the ellipse |
| {x_1^2\over a^2}+{y_1^2\over b^2} – 1 = 0 | on the ellipse |
| {x_1^2\over a^2}+{y_1^2\over b^2} – 1 < 0 | inside the ellipse |
Auxiliary Circle
The ellipse {x^2\over a^2}+{y^2\over b^2} = 1, (b < a), become x2 + y2 = a2, if b = a.
This is called the auxiliary circle of the ellipse i.e. the circle described on the major axis of an ellipse as diameter is called auxiliary circle.
