Advertisements
Advertisements
Question
Find the equation of the hyperbola referred to its principal axes:
whose foci are at (±2, 0) and eccentricity `3/2`
Solution
Let the required equation of hyperbola be
`x^2/"a"^2 - y^2/"b"^2` = 1. ...(i)
Given, eccentricity(e) = `3/2`
Co-ordinates of foci are (±ae, 0).
Given co-ordinates of foci are (±2, 0).
∴ ae = 2
∴ `"a"(3/2)` = 2
∴ a = `4/3`
∴ a2 = `16/9`
Now, b2 = a2(e2 – 1)
∴ b2 = a2e2 – a2
= `4 - 16/9`
= `20/9`
∴ The required equation of hyperbola is
`x^2/((16/9)) - y^2/((20/9))` = 1, i.e., `(9x^2)/16 - (9y^2)/20` = 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:
16x2 – 9y2 = 144
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^2/9` = 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).
Find the equation of the hyperbola referred to its principal axes:
whose distance between directrices is `8/3` and eccentricity is `3/2`
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:
whose vertices are (± 7, 0) and end points of conjugate axis are (0, ±3)
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^2/144 - 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:
Eccentricity of the hyperbola 16x2 − 3y2 − 32x − 12y − 44 = 0 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
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 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
If P(x1, y1) is a point on the hyperbola x2 - y2 = a2, then SP. S'P = ______.
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 ______.
(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 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 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 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?
