Advertisements
Advertisements
Question
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
Solution
Given equation of the hyperbola is `x^2/"a"^2 - y^2/"b"^2` = 1.
Let θ1 and θ2 be the inclinations.
m1 = tan θ1, m2 = tan θ2
Let P(x1, y1) be a point on the hyperbola
Equation of a tangent with slope ‘m’ to the hyperbola
`x^2/"a"^2 - y^2/"b"^2` = 1 is y = `"m"x ± sqrt("a"^2"m"^2 - "b"^2)`
This tangent passes through P(x1, y1).
∴ y1 = `"m"x_1 ± sqrt("a"^2"m"^2 - "b"^2)`
∴ (y1 – mx1)2 = a2m2 – b2
∴ (`"x"_1^2` – a2)m2 – 2x1y1m + (`"y"_1^2` + b2) = 0 …(i)
This is a quadratic equation in ‘m’.
It has two roots say m1 and m2, which are the slopes of two tangents drawn from P.
∴ m1 + m2 = `(2"x"_1"y"_1)/("x"_1^2-"a"^2)`
Since tan θ1 + tan θ2 = k,
`(2"x"_1"y"_1)/("x"_1^2-"a"^2)` = k
∴ P(x1, y1) moves on the curve whose equation is k(x2 – a2) = 2xy.
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:
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).
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 length of conjugate axis 6
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 transverse axis is 8 and distance between foci is 10
Find the equation of the tangent to the hyperbola:
9x2 – 16y2 = 144 at the point L of latus rectum in the first quadrant
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
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 equations of the tangents to the hyperbola 3x2 − y2 = 48 which are perpendicular to the line x + 2y − 7 = 0
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 ______.
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 ______.
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?
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 ______.
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 ______.
