Advertisements
Advertisements
Question
Find the equation of the hyperbola referred to its principal axes:
whose distance between directrices is `8/3` and eccentricity is `3/2`
Solution
Let the required equation of hyperbola be `x^2/"a"^2 - y^2/"b"^2` = 1
Given, eccentricity (e) = `3/2`
Distance between directrices = `(2"a")/"e"`
Given, distance between directrices = `8/3`
∴ `(2"a")/"e" = 8/3`
∴ `(2"a")/(3/2) = 8/3`
∴ `(4"a")/3 = 8/3`
∴ a = 2
∴ a2 = 4
∴ Now, b2 = a2(e2 – 1)
∴ b2 = `4[(3/2)^2 - 1]`
= `4(9/4 - 1)`
= `4(5/4)`
∴ b2 = 5
∴ The required equation of hyperbola is `x^2/4 - y^2/5` = 1.
APPEARS IN
RELATED QUESTIONS
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:
`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^2/100 - y^2/25` = + 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 equation of the hyperbola with centre at the origin, length of conjugate axis 10 and one of the foci (–7, 0).
Find the equation of the hyperbola referred to its principal axes:
whose distance between foci is 10 and length of conjugate axis 6
Find the equation of the hyperbola referred to its principal axes:
whose length of conjugate axis = 12 and passing through (1, – 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 and conjugate axis are 6 and 9 respectively
Find the equation of the tangent to the hyperbola:
3x2 – y2 = 4 at the point `(2, 2sqrt(2))`
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
Show that the line 3x – 4y + 10 = 0 is tangent till the hyperbola x2 – 4y2 = 20. Also find the point of contact
Select the correct option from the given alternatives:
Eccentricity of the hyperbola 16x2 − 3y2 − 32x − 12y − 44 = 0 is
Select the correct option from the given alternatives:
The foci of hyperbola 4x2 − 9y2 − 36 = 0 are
Answer the following:
Find the equation of the hyperbola in the standard form if Length of conjugate axis is 5 and distance between foci is 13.
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 7x2 − 3y2 = 51 at (−3, −2)
Answer the following:
Show that the line 2x − y = 4 touches the hyperbola 4x2 − 3y2 = 24. Find the point of contact
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
If P(x1, y1) is a point on the hyperbola x2 - y2 = a2, then SP. S'P = ______.
The eccentricity of the hyperbola 25x2 - 9y2 = 225 is ______.
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 ______.
(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 ______.
Let the hyperbola H : `x^2/a^2 - y^2/b^2` = 1 pass `(2sqrt(2), -2sqrt(2))`. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?
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 ______.
