Advertisements
Advertisements
प्रश्न
Write the distance between the directrices of the hyperbola x = 8 sec θ, y = 8 tan θ.
उत्तर
We have: \[x = 8\sec\theta, y = 8\tan\theta\]
On squaring and subtracting, we get:
\[ x^2 - y^2 = 64 \sec^2 \theta - 64 \tan^2 \theta\]
\[ \Rightarrow x^2 - y^2 = 64\]
\[ \Rightarrow \frac{x^2}{64} - \frac{y^2}{64} = 1\]
∴ a = b = 8
Distance between the directrices of hyperbola is \[\frac{2 a^2}{\sqrt{a^2 + b^2}}\].
\[\Rightarrow \frac{2 \times 64}{\sqrt{64 + 64}}\]
\[ = \frac{128}{8\sqrt{2}}\]
\[ = \frac{16}{\sqrt{2}}\]
\[ = 8\sqrt{2}\]
APPEARS IN
संबंधित प्रश्न
Find the equation of the hyperbola satisfying the given conditions:
Foci (±5, 0), the transverse axis is of length 8.
Find the equation of the hyperbola satisfying the given conditions:
Foci `(0, +- sqrt10)`, passing through (2, 3)
The equation of the directrix of a hyperbola is x − y + 3 = 0. Its focus is (−1, 1) and eccentricity 3. Find the equation of the hyperbola.
Find the equation of the hyperbola whose focus is (1, 1) directrix is 2x + y = 1 and eccentricity = \[\sqrt{3}\].
Find the equation of the hyperbola whose focus is (a, 0), directrix is 2x − y + a = 0 and eccentricity = \[\frac{4}{3}\].
Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .
9x2 − 16y2 = 144
Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .
16x2 − 9y2 = −144
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the distance between the foci = 16 and eccentricity = \[\sqrt{2}\].
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the conjugate axis is 5 and the distance between foci = 13 .
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the conjugate axis is 7 and passes through the point (3, −2).
Find the equation of the hyperbola whose vertices are (−8, −1) and (16, −1) and focus is (17, −1).
Find the equation of the hyperbola whose vertices are at (0 ± 7) and foci at \[\left( 0, \pm \frac{28}{3} \right)\] .
Find the equation of the hyperbola whose foci at (± 2, 0) and eccentricity is 3/2.
Find the equation of the hyperboala whose focus is at (5, 2), vertex at (4, 2) and centre at (3, 2).
Find the equation of the hyperbola satisfying the given condition :
foci (0, ± 13), conjugate axis = 24
Show that the set of all points such that the difference of their distances from (4, 0) and (− 4,0) is always equal to 2 represents a hyperbola.
Write the equation of the hyperbola whose vertices are (± 3, 0) and foci at (± 5, 0).
Equation of the hyperbola whose vertices are (± 3, 0) and foci at (± 5, 0), is
The equation of the hyperbola whose foci are (6, 4) and (−4, 4) and eccentricity 2, is
The foci of the hyperbola 2x2 − 3y2 = 5 are
The equation of the hyperbola whose centre is (6, 2) one focus is (4, 2) and of eccentricity 2 is
Find the equation of the hyperbola whose vertices are (± 6, 0) and one of the directrices is x = 4.
The length of the transverse axis along x-axis with centre at origin of a hyperbola is 7 and it passes through the point (5, –2). The equation of the hyperbola is ______.
If the distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`, then obtain the equation of the hyperbola.
Show that the set of all points such that the difference of their distances from (4, 0) and (– 4, 0) is always equal to 2 represent a hyperbola.
Find the equation of the hyperbola with foci `(0, +- sqrt(10))`, passing through (2, 3)
The equation of the hyperbola with vertices at (0, ± 6) and eccentricity `5/3` is ______ and its foci are ______.
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is ______.
The distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`. Its equation is ______.
