Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let the required equation of hyperbola be
`x^2/"a"^2 - y^2/"b"^2` = 1
Length of conjugate axis = 2b
Given, length of conjugate axis = 5
∴ 2b = 5
∴ b = `5/2`
∴ b2 = `25/4`
Distance between foci = 2ae
Given, distance between foci = 13
∴ 2ae = 13
∴ ae = `13/2`
∴ a2e2 = `169/4`
∴ Now, b2 = a2(e2 – 1)
∴ b2 = a2e2 – a2
∴ `25/4 = 169/4` – a2
∴ a2 = `169/4 - 25/4`
∴ a2 = `144/4` = 36
∴ The required equation of hyperbola is
`x^2/36 - y^2/(25/4)` = 1,
i.e., `x^2/36 - (4y^2)/25` = 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:
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^2/9` = 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:
`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 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 distance between foci is 10 and eccentricity `5/2`
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 transverse and conjugate axis are 6 and 9 respectively
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:
3x2 – 4y2 = 12 at the point (4, 3)
Find the equation of the tangent to the hyperbola:
`x^2/144 - y^2/25` = 1 at the point whose eccentric angle is `pi/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:
If the line 2x − y = 4 touches the hyperbola 4x2 − 3y2 = 24, the point of contact is
Answer the following:
Find the equation of the hyperbola in the standard form if length of the conjugate axis is 3 and distance between the foci is 5.
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:
Show that the line 2x − y = 4 touches the hyperbola 4x2 − 3y2 = 24. Find the point of contact
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
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 ______.
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 asymptotes of the hyperbola xy = hx + ky are ______.
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 equation of conjugate axis for the hyperbola `(x + y + 1)^2/4 - (x - y + 2)^2/9` = 1 is ______.
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 ______.
The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x2sec2α – y2cosec2α = 1, `α∈(0, π/4)` are ______.
Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola `x^2/"a"^2 - "y"^2/"b"^2` = 1. Let e' and l' respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If e2 = `11/14"l'"` and (e')2 = `11/8"l"^'` then the value of 77a + 44b 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 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 ______.
