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If any two chords be drawn through two p...

If any two chords be drawn through two points on the major axis of the ellipse `x^2/a^2+y^2/b^2=1` equidistant from the centre prove that `tan""alpha/2 tan""beta/2 tan""gamma/2=1` where `alpha,beta,gamma,delta` are the eccentricity angles of the extremities of the chord.

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Prove that the chord joining points P(alpha) and Q (beta) on the ellipse x^2/a^2+y^2/b^2=1 subtends a right angle at the vertex A(a,0) then tan""alpha/2 tan""beta/2=(-b^2)/a^2 .

Find the area of the triangle formed by three points on the ellipse x^2/a^2+y^2/b^2=1 whose eccentric angles are alpha, beta and gamma.

Knowledge Check

  • If the chord joining two points whose eccentric angles are alpha and beta cut the major axis of an ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at a distance from the centre then tan (alpha)/(2).tan (beta)/(2) =

    A
    `(c+a)/(c-a)`
    B
    `(c-a)/(c+a)`
    C
    `(c )/(a )`
    D
    c+a

  • If alpha,beta are the ends of a focal chord of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 then its eccentricity e is

    A
    `(1+tan(alpha)/(2)tan(beta)/(2))/(1-tan (alpha)/(2)tan (beta)/(2))`
    B
    `(1-tan(alpha)/(2)tan(beta)/(2))/(1+tan (alpha)/(2)tan (beta)/(2))`
    C
    `(tan(alpha)/(2)tan(beta)/(2)+1)/(tan(alpha)/(2)tan (beta)/(2)-1)`
    D
    `(tan(alpha)/(2)tan(beta)/(2)-1)/(tan (alpha)/(2)tan (beta)/(2)+1)`

  • If alpha, beta are the eccentric angles of the extremities of a focal chord of the ellipse x^(2)/16+y^(2)/9, " then "tan""alpha/2tan""beta/2=

    A
    `(sqrt(5)+4)/(sqrt(5)-4)`
    B
    `9/23`
    C
    `(sqrt(5)-4)/(sqrt(5)+4)`
    D
    `(8sqrt(7)-23)/9`

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    IF alpha,beta are the eccentric angles of the extremities of a focal chord of the ellipse x^2/a^2+y^2/b^2=1 . Then show that e cos""(alpha+beta)/2=cos""(alpha-beta)/2

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