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
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
i. Length of transverse axis = 2a = 2(4) = 8
Length of conjugate axis = 2b = 2(4) = 8
ii. 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)`
iii. Equations of the directrices are x = `± "a"/"e"`.
∴ x = `± 4/sqrt(2)`
∴ x = `±2sqrt(2)`
iv. Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/4`
= 8
v. Distance between foci = 2ae = `2(4)(sqrt(2)) = 8sqrt(2)`
vi. Distance between directrices = `(2"a")/"e"`
= `(2(4))/sqrt(2)`
= `4sqrt(2)`.
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:
5y2 = 24x
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 (1, –6)
Find the focal distance of a point on the parabola y2 = 16x whose ordinate is 2 times the abscissa
Find coordinates of the point on the parabola. Also, find focal distance.
y2 = 12x whose parameter is `1/3`
Find coordinates of the point on the parabola. Also, find focal distance.
2y2 = 7x whose parameter is –2
For the parabola y2 = 4x, find the coordinate of the point whose focal distance is 17
Find length of latus rectum of the parabola y2 = 4ax passing through the point (2, –6)
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find its focus.
Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y = x2 – 2x + 3
If the tangent drawn from the point (–6, 9) to the parabola y2 = kx are perpendicular to each other, find k
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 area of the triangle formed by the line joining the vertex of the parabola x2 = 12y to the endpoints of its latus rectum is _________
Select the correct option from the given alternatives:
The equation of the parabola having (2, 4) and (2, –4) as endpoints of its latus rectum is _________
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:
A line touches the circle x2 + y2 = 2 and the parabola y2 = 8x. Show that its equation is y = ± (x + 2).
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
The area of the triangle formed by the lines joining vertex of the parabola x2 = 12y to the extremities of its latus rectum is ______.
The equation of the directrix of the parabola 3x2 = 16y is ________.
Let P: y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of `π/4` with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is ______.
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose directrix is ______.
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 ______.
Which of the following are not parametric coordinates of any point on the parabola y2 = 4ax?
The equation to the line touching both the parabolas y2 = 4x and x2 = –32y is ______.
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 ______.
Through the vertex O of parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another, where P and Q are points on the parabola. If the locus of middle point of PQ is y2 = 2(x – l), then value of l is ______.
Let a variable point A be lying on the directrix of parabola y2 = 4ax (a > 0). Tangents AB and AC are drawn to the curve where B and C are points of contact of tangents. The locus of centroid of ΔABC is a conic whose length of latus rectum is λ, then `λ/"a"` is equal to ______.
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 ______.
The cartesian co-ordinates of the point on the parabola y2 = –16x, whose parameter is `1/2`, are ______.
