Find The Coordinates Of The Foci And The Equation Of The Directrices Of The Rectangular Hyperbola xy = c2.

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Question

Find the coordinates of the foci and the equation of the directrices of the rectangular hyperbola xy = c².

medium

Solution

Correct option is

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

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SIMILAR QUESTIONS

Q1

Find the equation and the length of the common tangents to hyperbola

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Q2

Find the locus of the foot of perpendicular from the centre upon any normal to the hyperbola .

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Q3

Find the locus of the mid-points of the chords of the hyperbola which subtend a right angle at the origin.

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Q4

Find the locus of the poles of normal chords of the hyperbola

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Q5

Find the condition for the lines Ax² + 2Hxy + By² = 0 to be conjugate diameters of .

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Q6

Find the asymptotes of the hyperbola xy – 3y – 2x = 0.

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Q7

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.

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Q8

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.

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Q9

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.

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Q10

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.

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