Find the coordinates of the foci and the equation of the directrices of the rectangular hyperbola xy = c2.
Reference to transverse and conjugate axes, as coordinate axes, the equation of hyperbola is
If e be the eccentricity, then
∴ Coordinates of foci are
Also its direction are
Now rotating the coordinate axes through an angle –π/4, the equation of hyperbola reduces to xy = c2 where c2 = a2/2, by putting for x and y.
∴ For foci, we get
Also the equations of the directrices reduce to
Find the equation and the length of the common tangents to hyperbola
Find the locus of the foot of perpendicular from the centre upon any normal to the hyperbola .
Find the locus of the mid-points of the chords of the hyperbola which subtend a right angle at the origin.
Find the locus of the poles of normal chords of the hyperbola
Find the condition for the lines Ax2 + 2Hxy + By2 = 0 to be conjugate diameters of .
Find the asymptotes of the hyperbola xy – 3y – 2x = 0.
A ray emanating from the point (5, 0) is incident on the hyperbola 9x2 – 16y2 = 144 at the point P with abscissa 8. Find the equation of the reflected ray after first reflection and point P lies in first quadrant.
The equations of the transverse and conjugate axes of a hyperbola are respectively 3x + 4y – 7 = 0, 4x – 3y + 8 = 0 and their respective lengths are 4 and 6. Find the equation of the hyperbola.
A, B, C are three points on the rectangular hyperbola xy = c2, find
1. The area of the triangle ABC
2. The area of the triangle formed by the tangents at A, B and C.
Find the equation of the hyperbola whose asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and which passes through the point
(1,–1 ). Find also the equation of the conjugate of the conjugate hyperbola.