Find the locus of the extremities of the latus recta of all ellipses having a given major axis 6a?

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Saurabh Ashok Pugaliya

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Let LSI be the latus rectum, C be the centre of the ellipse and the coordinates of L be (x, y) then x = CS = 3 ae ...... (1) And y = SL = b2/3a = (9a2(1 - e2))/3a = 3a (1 - e2) ...... (2) Eliminating the variable 'e' from (1) and (2) we get the locus of L. Hence putting the value of e from (1) and (2),...
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Let LSI be the latus rectum, C be the centre of the ellipse and the coordinates of L be (x, y) then x = CS = 3 ae ...... (1) And y = SL = b2/3a = (9a2(1 - e2))/3a = 3a (1 - e2) ...... (2) Eliminating the variable 'e' from (1) and (2) we get the locus of L. Hence putting the value of e from (1) and (2), we get:- y = 3a(1-x2/9a2) x2 = 9a2 - 3ay x2 = 3a(3a - y), which is clearly a parabola. Similarly, we can show that the locus of L' is x2 = 3ay(y + 3a) which is again a parabola. read less
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IIT Bombay

Co-ordinate of extreme point of latus rectum would b (c,y).You can find ‘y’ by taking ‘x’ as ‘c’ in general equation,which comes out to b2/a.So the co-ordinate becomes {(a2-b2)1/2,b2/a}.Eliminating ‘b’ you will get a2-x2=ay.
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