Advertisements
Advertisements
प्रश्न
Answer the following:
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) Distance between foci
(vi) distance between directrices of the curve
`x^2/144 - y^2/25` = 1
Advertisements
उत्तर
Given equation of the hyperbola `x^2/144 - y^2/25` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1
a2 = 144 and b2 = 25
∴ a = 12 and b = 5
i. Length of transverse axis = 2a = 2(12) = 24
Length of conjugate axis = 2b = 2(5) = 10
∴ lengths of the principal axes are 24 and 10.
ii. b2 = a2(e2 – 1)
∴ 25 = 144 (e2 – 1)
∴ `25/144` = e2 – 1
∴ e2 = `1 + 25/144`
∴ e2 = `169/144`
∴ e = `13/12` ...[∵ e > 1]
Co-ordinates of foci are S(ae, 0) and S'(–ae, 0)
i.e., `"S"(12(13/12),0)` and `"S'"(-12(13/12),0)`
i.e., S(13, 0) and S'(–13, 0)
iii. Equations of the directrices are x = `± "a"/"e"`
i.e., x = `± 12/((13/12))`, i.e., x = `± 144/13`
iv. Length of latus rectum = `(2"b"^2)/"a"`
= `(2(25))/12`
= `25/6`
v. Distance between foci = 2ae = `2(12)(13/12)` = 26
vi. Distance between directrices = `(2"a")/"e"`
= `(2(12))/((13/12))`
= `288/13`
APPEARS IN
संबंधित प्रश्न
Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:
3x2 = 8y
Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:
x2 = –8y
Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:
3y2 = –16x
Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (3, 4)
Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (2, 3)
For the parabola 3y2 = 16x, find the parameter of the point (3, – 4).
For the parabola 3y2 = 16x, find the parameter of the point (27, –12).
Find coordinates of the point on the parabola. Also, find focal distance.
2y2 = 7x whose parameter is –2
Find the area of the triangle formed by the line joining the vertex of the parabola x2 = 12y to the end points of latus rectum.
Find the equation of tangent to the parabola y2 = 12x from the point (2, 5)
Find the equation of tangent to the parabola y2 = 36x from the point (2, 9)
Find the equation of the locus of a point, the tangents from which to the parabola y2 = 18x are such that some of their slopes is –3
The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the roadway and are 200 meters apart. If the cable is 5 meters above the roadway at the centre of the bridge, find the length of the vertical supporting cable 30 meters from the centre.
A circle whose centre is (4, –1) passes through the focus of the parabola x2 + 16y = 0.
Show that the circle touches the directrix of the parabola.
Select the correct option from the given alternatives:
The line y = mx + 1 is a tangent to the parabola y2 = 4x, if m is _______
Select the correct option from the given alternatives:
If the focus of the parabola is (0, –3) its directrix is y = 3 then its equation is
Select the correct option from the given alternatives:
Equation of the parabola with vertex at the origin and directrix x + 8 = 0 is __________
Answer the following:
For the following parabola, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum:
5x2 = 24y
Answer the following:
Find the Cartesian coordinates of the point on the parabola y2 = 12x whose parameter is −3
Answer the following:
Find the equation of the tangent to the parabola y2 = 9x at the point (4, −6) on it
Answer the following:
Find the equation of the tangent to the parabola y2 = 8x at t = 1 on it
Answer the following:
Show that the two tangents drawn to the parabola y2 = 24x from the point (−6, 9) are at the right angle
Answer the following:
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) Distance between foci
(vi) distance between directrices of the curve
x2 − y2 = 16
The length of latus-rectum of the parabola x2 + 2y = 8x - 7 is ______.
The area of the triangle formed by the lines joining vertex of the parabola x2 = 12y to the extremities of its latus rectum is ______.
If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (–30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is ______.
If the line `y - sqrt(3)x + 3` = 0 cuts the parabola y2 = x + 2 at A and B, then PA. PB is equal to `("where coordinates of P are" (sqrt(3), 0))` ______.
If the normal at the point (1, 2) on the parabola y2 = 4x meets the parabola again at the point (t2, 2t), then t is equal to ______.
If the vertex = (2, 0) and the extremities of the latus rectum are (3, 2) and (3, –2) then the equation of the parabola is ______.
The equation of the parabola whose vertex and focus are on the positive side of the x-axis at distances a and b respectively from the origin is ______.
The equation of the line touching both the parabolas y2 = x and x2 = y is ______.
A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola y = `(x - 1/4)^2 + α`, where α > 0. Then (4α – 8)2 is equal to ______.
If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21, then the length of its latus rectum is ______.
The cartesian co-ordinates of the point on the parabola y2 = –16x, whose parameter is `1/2`, are ______.
