Advertisements
Advertisements
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:
x2 – y2 = 16
Advertisements
उत्तर
Given equation of the hyperbola is x2 – y2 = 16
∴ `x^2/16 - y^2/16` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get
a2 = 16 and b2 = 16
∴ a = 4 and b = 4
Length of transverse axis = 2a = 2(4) = 8
Length of conjugate axis = 2b = 2(4) = 8
We know that
e =`sqrt("a"^2 + "b"^2)/"a"`
= `sqrt(16 + 16)/4`
= `sqrt(32)/4`
= `(4sqrt(2))/4`
= `sqrt(2)`
Co-ordinates of foci are S(ae, 0) and S'(– ae, 0),
i.e., `"S"(4sqrt(2), 0)` and `"S""'"(-4 sqrt(2), 0)`
Equations of the directrices are x = `± "a"/"e"`.
∴ x = `± 4/sqrt(2)`
∴ x = `±2sqrt(2)`
Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/4`
= 8.
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:
16x2 – 9y2 = 144
Find the equation of the hyperbola with centre at the origin, length of conjugate axis 10 and one of the foci (–7, 0).
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:
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 foci are at (±2, 0) and eccentricity `3/2`
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
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 `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:
If the line 2x − y = 4 touches the hyperbola 4x2 − 3y2 = 24, the point of contact is
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 Length of conjugate axis is 5 and distance between foci is 13.
Answer the following:
Find the equation of the tangent to the hyperbola 7x2 − 3y2 = 51 at (−3, −2)
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
The eccentricity of the hyperbola 25x2 - 9y2 = 225 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.
The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point of intersection of the lines, 7x + 13y – 87 = 0 and 5x – 8y + 7 = 0, the latus rectum is `32sqrt(2)/5`. The value of `(asqrt(2) + b)` will be ______.
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 ______.
