Advertisements
Advertisements
प्रश्न
If e and e' are the eccentricities of a hyperbola and its conjugate hyperbola respectively, prove that `1/"e"^2 + 1/("e""'")^2` = 1
Advertisements
उत्तर
Let e be the eccentricity of the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1
∴ e = `sqrt("a"^2 + "b"^2)/"a"`
∴ e2 = `("a"^2 + "b"^2)/"a"^2`
Let e' be the eccentricity of the conjugate hyperbola `y^2/"b"^2 - x^2/"a"^2` = 1
∴ e' = `sqrt("b"^2 + "a"^2)/"b"`
∴ `"e'"^2` = `("a"^2 + "b"^2)/"b"^2`
∴ `1/"e"^2 + 1/"e'"^2 = "a"^2/("a"^2 + "b"^2) + "b"^2/("a"^2 + "b"^2)`
= `("a"^2 + "b"^2)/("a"^2 + "b"^2)`
= 1
APPEARS IN
संबंधित प्रश्न
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
`x^2/25 - y^2/16` = 1
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
`x^2/25 - y^2/16` = – 1
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
3x2 – y2 = 4
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
x2 – y2 = 16
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
`y^2/25 - x^2/144` = 1
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
x = 2 sec θ, y = `2sqrt(3) tan theta`
Find the eccentricity of the hyperbola, which is conjugate to the hyperbola x2 – 3y2 = 3
Find the equation of the hyperbola referred to its principal axes:
whose distance between foci is 10 and eccentricity `5/2`
Find the equation of the hyperbola referred to its principal axes:
which passes through the points (6, 9) and (3, 0)
Find the equation of the hyperbola referred to its principal axes:
whose vertices are (± 7, 0) and end points of conjugate axis are (0, ±3)
Find the equation of the hyperbola referred to its principal axes:
whose length of transverse axis is 8 and distance between foci is 10
Find the equation of the tangent to the hyperbola:
`x^2/16 - y^2/9` = 1 at the point in a first quadratures whose ordinate is 3
Find the equations of the tangents to the hyperbola `x^2/25 - y^2/9` = 1 making equal intercepts on the co-ordinate axes
Find the equations of the tangents to the hyperbola 5x2 – 4y2 = 20 which are parallel to the line 3x + 2y + 12 = 0
Select the correct option from the given alternatives:
The foci of hyperbola 4x2 − 9y2 − 36 = 0 are
Answer the following:
For the hyperbola `x^2/100−y^2/25` = 1, prove that SA. S'A = 25, where S and S' are the foci and A is the vertex
Answer the following:
Find the equation of the hyperbola in the standard form if eccentricity is `3/2` and distance between foci is 12.
Answer the following:
Find the equation of the tangent to the hyperbola x = 3 secθ, y = 5 tanθ at θ = `pi/3`
Answer the following:
Find the equations of the tangents to the hyperbola 3x2 − y2 = 48 which are perpendicular to the line x + 2y − 7 = 0
Answer the following:
Two tangents to the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 make angles θ1, θ2, with the transverse axis. Find the locus of their point of intersection if tan θ1 + tan θ2 = k
The eccentricity of the hyperbola 25x2 - 9y2 = 225 is ______.
Let H: `x^2/a^2 - y^2/b^2` = 1, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is `4(2sqrt(2) + sqrt(14))`. If the eccentricity H is `sqrt(11)/2`, then the value of a2 + 2b2 is equal to ______.
The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, `x^2/9 - y^2/16` = 1 is ______.
A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola `x^2/4 - y^2/2` = 1 at the point (x1, y1). Then `x_1^2 + 5y_1^2` is equal to ______.
The asymptotes of the hyperbola xy = hx + ky are ______.
The foci of a hyperbola coincide with the foci of the ellipse `x^2/25 + y^2/9` = 1. Find the equation of the hyperbola, if its eccentricity is 2.
(x – 1)2 + (y – 2)2 = `(3(2x + 3y + 2)^2)/13`represents hyperbola whose eccentricity is ______.
Parametric form of the hyperbola `x^2/4 - y^2/9` = –1 is ______.
The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point of intersection of the lines `x - 3sqrt(5)y` = 0 and `sqrt(5)x - 2y` = 13 and the length of its latus rectum is `4/3` units. The coordinates of its focus are ______.
The locus of the mid-point of the chords of the hyperbola `(x^2/a^2) - (y^2/b^2)` = 1 passing through a fixed point (α, β) is a hyperbola with centre at `(α/2, β/2)` It equation is ______.
If the radii of director circles of `x^2/a^2 + y^2/b^2` = 1 and `x^2/a^2 - y^2/b^2` = (a > b) are 2r and r respectively, then `e_2^2/e_1^2` is equal to ______.
(where e1, e2 are their eccentricities respectively)
Let e1 and e2 be the eccentricities of the ellipse, `x^2/25 + y^2/b^2` = 1 (b < 5) and the hyperbola, `x^2/16 - y^2/b^2` = 1 respectively satisfying e1e2 = 1. If α and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair (α, β) is equal to ______.
The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point `(3sqrt(5), 1)` and the length of its latus rectum is `4/3` units. The length of the conjugate axis is ______.
The eccentricity of the hyperbola x2 – 3y2 = 2x + 8 is ______.
