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Directrix of ellipse

2022-06-11 07:12:00 【lihongtao8209】

As shown in the figure above , It is known that ： The ellipse O The standard equation for is ： $\frac{x^2}{16} +\frac{y^2}{4} =1$ , $F_{1}$ 、 $F_{2}$ It's an ellipse O The two focuses of . Line segment AD In the ellipse O On the left horizontal line of . $F_{1}P$ It's an ellipse O A focal radius of . Find the ellipse O The collimation equation of .

In the unified definition of conic : The distance from a fixed point to a fixed line Than Constant e(e>0) Point of The trajectory , It is called conic curve . And this straight line is called the collimation line .0<e<1 when , The trajectory is an ellipse ; e=1 when , The trajectory is a parabola ; e>1 when , The trajectory is hyperbolic . The parabola is the same as p It's about .

According to the above definition ：

Made a moving point of ellipse P do PA Perpendicular to AD. because AD On the left horizontal line , therefore PA Perpendicular to the left horizontal line . The length of L.
Made a moving point of ellipse P do PB Perpendicular to x Axis .
$e=\frac{c}{a}$

Sum up , A moving point on an ellipse P To left focus $F_{1}$ Distance of | $|PF_{1}|$ And the moving point P The distance to the left horizontal line L The ratio is $\frac{a}{c}$ .