Advertisements
Advertisements
Question
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:
16x2 – 9y2 = 144
Advertisements
Solution
Given equation of the hyperbola is 16x2 – 9y2 = 144
∴ `x^2/9 - y^2/16` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get,
a2 = 9 and b2 = 16
a = 3 and b = 4
(1) Length of transverse axis = 2a = 2(3) = 6
(2) Length of conjugate axis = 2b = 2(4) = 8
(3) Eccentricity = e = `sqrt("a"^2 + "b"^2)/"a"`
= `sqrt(9 + 16)/3`
= `sqrt(25)/3`
= `5/3`
(4) Co-ordinates of foci are S(ae, 0) and S'(−ae, 0),
i.e., `"S"(3(5/3),0)` and `"S'"(-3(5/3),0)`,
i.e., S(5, 0) and S'(−5, 0)
(5) Equations of the directrices are x = `±"a"/"e"`
∴ x = `± 3/((5/3))`
∴ x = `± 9/5`
(6) Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/3`
= `32/3`
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:
21x2 – 4y2 = 84
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
If e and e' are the eccentricities of a hyperbola and its conjugate hyperbola respectively, prove that `1/"e"^2 + 1/("e""'")^2` = 1
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:
`x^2/16 - y^2/9` = 1 at the point in a first quadratures whose ordinate is 3
Find the equation of the tangent to the hyperbola:
9x2 – 16y2 = 144 at the point L of latus rectum in the first quadrant
Show that the line 3x – 4y + 10 = 0 is tangent till the hyperbola x2 – 4y2 = 20. Also find the point of contact
If the 3x – 4y = k touches the hyperbola `x^2/5 - (4y^2)/5` = 1 then find the value of k
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 eccentricity of rectangular hyperbola is
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 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 equation of the tangent to the hyperbola `x^2/25 − y^2/16` = 1 at P(30°)
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
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 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.
Parametric form of the hyperbola `x^2/4 - y^2/9` = –1 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 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 ______.
For the Hyperbola `x^2/(cos^2α) - y^2/(sin^2α)` = 1, which of the following remains constant when α varies = ?
The eccentricity of the hyperbola x2 – 3y2 = 2x + 8 is ______.
