Ellipse
An ellipse is the set of points in the plane, the sum of whose distance from two fixed points is a given positive constant that is greater than the distance between the two fixed point..
Equation of Ellipse
Equation of ellipse is given by (x - h) 2 / a 2 + (y - k) 2 / b 2 = 1 . Where h, k a and b are real number..
Equation of tangent and normal to an ellipse
Equation of tangent and normal to an ellipse is important with respect to the angle that is made by the lines inside the ellipse as we have seen in auxiliary angl..
General Equation of an Ellipse
- Let e be the eccentricity of an ellipse whose focus is F(p,q) and the equation of whose directrix is ax+by+c=0 Let P(x,y) be a point on the ellipse and PM be the perpendicular on directrix .Then by definition of ellipse PF/PM = e PF 2 = e 2 PM 2 (x-p) 2 + (..
Choose an equation for the ellipse shown.
Choose an equation for the ellipse shown. => 4 x 2 + y 2 = 4 or x 2 + 2 y 2 = 2 or 2 x 2 + y 2 = 2 or x 2 + 4 y 2 = 4..
Parametric Representation of Ellipse
If the Point of contacts on the ellipse ,when the lines or the circles in touch with the ellipse ,we call it as parametric representation of ellipse..
Verticle Form of an Ellipse
Major axis in verticle form of ellipse will be along y axis and the minor axis will be x axis. To derive at the equation of verticle form of ellipse we need to get the points of directrices and length of latus rectu..
Example Problems on Ellipse
Answer - Let the given ellipse be (x 2 /a 2 )+(y 2 /b 2 )=1 Let P(x1,y1) be one end of a diameter of the ellipse .Then another end is (-x1,-y1..
Verticle Form of an Ellipse
- In this case the major and minor axis of the ellipse along y-axis and x-axis respectively . AA’ =2b and BB’=2a .The foci F and F’ are (0,be)respectively . The directrices DZ and DZ are given by y=b/e and y= -b/e respectively.Length of latus rectum 2b 2 /a The equa..
- In this case the major and minor axis of the ellipse along y-axis and x-axis respectively . AA’ =2b and BB’=2a .The foci F and F’ are (0,be)respectively . The directrices DZ and DZ are given by y=b/e and y= -b/e respectively.Length of latus rectum 2b 2 /a The equa..Parametric Representation of Ellipse
Proof - Let QPN be perpendicular to the major axis and O be the centre of the ellipse. Let NOQ=Φ,Then X = ON=OQcosΦ = acosΦ.Since p(x,y) lies on the ellipse (x 2 /a 2 )+(y 2 /b 2 ) = 1 and since x=acosΦ we have[( a 2 cos 2 Φ/a 2 )/a 2 ]+[y 2 /b 2 ]=1 or y 2 /..
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