The line segment connecting any two points on the ellipse is called the chord, and the chord passing through the center of the ellipse is called the diameter.The locus AB of the midpoint of the chord parallel to the diameter DE and the diameter DE are mutually conjugate diameters.Similarly, the conjugate diameter of a hyperbola can be defined.
Chinese name
Conjugate diameter
Ellipse
Countless pairs
Perpendicular to each other
Is the major axis and minor axis of the ellipse
Diameter
Arbitrary
Definition
The line segment connecting any two points on the ellipse is called the chord, and the chord passing through the center of the ellipse is called the diameter.The locus AB of the midpoint of the chord parallel to the diameter DE and the diameter DE are mutually conjugate diameters
An ellipse, whose center is O, passes through O and makes a diameter CD, then makes a parallel chord EF of CD, takes the midpoint M of EF, connects OM to get another straight diameter AB of the ellipse, then AB and CD are called a pair of conjugate diameters of the ellipse, and EF is the conjugate chord of the diameter AB.Therefore, any diameter of the ellipse must bisect its conjugate chord.Since the AB diameter above is arbitrarily chosen, there are countless pairs of conjugate diameters of the ellipse.
When a pair of conjugate diameters are perpendicular to each other, they are the major axis and minor axis of the ellipse.
Figure 1
Elliptical conjugate diameter
Announce
edit
brief introduction
Pass the ellipseE：
(a>b>0) ofstringACIs called the diameter of the ellipse.[1]
If P is an ellipseESuperior toA、CPoints, there are
。
If the slope product of the two diameters of the ellipse is
, we call these two diameters the conjugate diameters of the ellipse.
In particular, if the slope of one diameter is 0 and the slope of the other diameter does not exist, the two diameters are also called conjugate diameters.[1-2]
nature
Nature I
set upAC，BDIs an ellipseEA pair of conjugate diameters of, ifA、BThe coordinates of two points are（xone，yone)、(xtwo，ytwo）, then
（1）
or
；[1]
（2）
；
（3）
；
（4）
；
（5）
。
Nature 2
set upAC，BDIs an ellipseEA pair of conjugate diameters of,PIs an ellipseEAt any point above, then
。
Nature 3
set upAC，BDIs an ellipseEA pair of conjugate diameters of,MIs a segmentABMidpoint of, rayOMIntersecting ellipseEAt pointP, then
。
Nature 4
set upAC，BDIs an ellipseEA pair of conjugate diameters of,PIs an ellipseEAny point above, passingPdoBD，ACParallel lines ofAC，BDtoM，N, then
。
Nature 5
set upAC，BDIs an ellipseEA pair of conjugate diameters of,PIs an ellipseEAny point above, passingPMake an ellipseETangents of intersect separatelyAC，BDExtender for onJ、I, respectively through PBD、ACParallel lines ofAC，BDtoM、N, then
