# Hyperbola - 5

**CONJUGATE HYPERBOLA**

The hyperbola whose transverse & conjugate axes are respectively the conjugate & transverse axes of the standard hyperbolais called the conjugate hyperbola of the standard given hyperbola. The conjugate hyperbola of is = - 1.

**SOME RESULTS OF CONJUGATE HYPERBOLA**(a) Length of Transverse axis = 2b

(b) Length of conjugate axis = 2a

© Foci = (0, ±be)

(d) Equation of Directrices = y = ± b/e

(e) Eccentricity = e =

(f) Length of Latus Rectum =

(g) Parametric co-ordinates = (a tan, b sec)

(h) Equation of transverse axis = x = 0

(i) Equation of conjugate axis = y = 0

__.__

**Illustration : 11**Find the length of transverse axis, conjugate axis eccentricity, coordinates of foci, vertices, length of latus rectum & equation of directrices of the hyperbola 9x

^{2}- 4y

^{2}= - 36.

Ans: The hyperbola is

9x

^{2}- 4y

^{2}= - 36

= - 1 i.e of the form

= - 1 where a = 2 & b = 3

Length of transverse axis = 2b = 6units

Length of conjugation axis = 2a = 4 units

Eccentricity =

Vertices = (0, ± b) = (0, ± 3)

Length of Latus Rectum = units

Equation of directrices = y = ± b/e

= y = ±

= y = ±

= y = ±

**RECTANGULAR HYPERBOLA DEFINITION**(1) A hyperbola whose asymptotes include aright angle is called a rectangular Hyperbola. THe general form of the equation of hyperbola is x

^{2}- y

^{2}= a

^{2}= b

^{2}.

Why ?

The asysmptotes of a hyperbola is

y = ±

If these are at right angles than

m

_{1}m

_{2}= - 1

= - 1

b

^{2}= a

^{2}

Hence the equation of ractangular hyperbola is = 1

x

^{2}- y

^{2}= a

^{2}

(2) If the length of transverse & conjugate axes of any hyperbola are equal, it is called as rectangular hyperbola.

Why ?

IF a = b, then = 1 become

= 1 x

^{2}- y

^{2}= a

^{2}

**RECTANGULAR HYPERBOLA****(xy = c**

^{2})The equation of Rectangular Hyperbola is x

^{2}- y

^{2}= a

^{2}

i.e. (x + y)(x - y) = a

^{2}

We know that x + y =0 & x - y = 0 are at 45

^{0}& 135

^{0}to the x axis. Now if we can rotate the axes through = - 45

^{0}without changing the origin.

So, we can replace (x, y) by

(x cos - y sin, x sin + y cos) i.e.

The equation x

^{2}- y

^{2}= a

^{2}becomes

= a

^{2}

xy =

Let = c

^{2}= any positive constant

Therefore xy = c

^{2}is the another form of Rectangular hyperbola.

**RESULTS OF RECTANGULAR HYPERBOLA**(a) The parametric coordinates are x = ct & y = c/t.

(b) Equation of chord joining t

_{1}& t

_{2}is

x + yt

_{1}t

_{2}= c(t

_{1}+ t

_{2})

© Equation of tangent at t is + yt = x.

(d) Point of intersection of tangents at ’t

_{1}’ & ’t

_{2}’ is .

(e) Equation of normal at ’t’ is

xt

^{3}- yt - ct

^{4}+ c = 0.

(f) Points of intersection of normals at ’t

_{1}’ & ’t

_{2}’ is

__.__

**Illustration : 12**Find the eccentricity of Rectangular Hyperbola.

Ans: For the hyperbola = 1; c =

Therefore for = 1 we have b = a

Hence eccentricity of Rectangular hyperbola is .

**Easy**1. For what value of k does = 1 represents a hyperbola ?

Ans: We know that the standard form of hyperbola can be = 1 or = - 1.

**CASE - I**= 1

As a

^{2}& b

^{2}are +ve quantities greater than zero simultaneously, therefore

3 + > 0 & 6 - > 0

> - 3 & < 6

Hence common solution is

- 3 < < 6

**CASE II**= - 1 or = 1

Here also - a

^{2}& - b

^{2}are -ve quantities.

Therefore

3 + < 0 & also 6 - < 0

< - 3 & > 6

There is no common solution.

Hence the only solution is

- 3 < < 6 or (- 3, 6)

Q.2. Find the value of c if the ellipse = 1 & x

^{2}- y

^{2}= c

^{2}cut at right angles ?

Ans: The curves cut at right angle means that the tangents at the point of intersection of two curves are at right angles. If m

_{1}is the slope of the tangent to ellipse & m

_{2}is the slope of the tangent to hyperbola, then

m

_{1}= …………………….. (i)

m

_{2}= ………………………………… (ii)

using (i) & (ii)

m

_{1}m

_{2}= - 1

= 1

But = 1 as (x

_{1}y

_{1}) lies on hyperbola

x

_{1}

^{2}=

^{9}⁄

_{2}………………………………………… (iii)

y

_{1}

^{2}=

^{4}⁄

_{2}……………………………………….. (iv)

Bur x

_{1}

^{2}- y

_{1}

^{2}= c

^{2}…………………………………… (v)

Using (iii) & (iv) in (v) we get

c

^{2}= c = ±

Q.3. Prove that if normal to the hyperbola xy = c

^{2}at any point t

_{1}meets the curve again at t

_{2}, then, t

_{1}

^{3}t

_{2}= - 1.

Ans: Equation of normal at point t

_{1}is

yt

_{1}- xt

_{1}

^{3}= c - ct

_{1}

^{4}

Now if this normal again meets at ’t

_{2}’, then (ct

_{2}, ) must satisfy the normal

t

_{1}- ct

_{2}t

_{1}

^{3}= c - ct

_{1}

^{4}

ct

_{1}

^{3}[t

_{2}- t

_{1}] = [t

_{1}- t

_{2}]

ct

_{1}

^{3}t

_{2}[t

_{2}- t

_{1}] + c(t

_{2}- t

_{1}) = 0

c(t

_{2}- t

_{1})(t

_{1}

^{3}+ 1) = 0

As t

_{2}t

_{1}& c > 0

t

_{1}

^{3}t

_{2}+ 1 = 0

t

_{1}

^{3}t

_{2}= - 1

Q.4. If (a sec, b tan) & (a sec, b tan) are the end points of a focal chord of the hyperbola = 1, then prove that tan tan = .

Ans: The equation of chord joining the points (a sec, b tan) & (a sec, b tan) is

It passes through focus (ae, 0), then

Applying componendo & dividendo we get

Q.5. Find the locus of poles of normal chords of the rectyangular hyperbola xy = c

^{2}.

Ans: Equation of normal at any t is

xt

^{3}- yt = c(t

^{4}- 1) …………………………………. (i)

Lert its ple be P(x

_{1}, y

_{1}) then the equation of polar is

xy

_{1}+ x

_{1}y = 2c

^{2}…………………………………………….. (ii)

Comparing the coefficients of equations (i) & (ii) as they represent same equation

t

^{2}= - …………………………………………………….. (iii)

& …………………………………………………. (iv)

From (iii) & (iv)

(x

_{1}

^{2}- y

_{1}

^{2})

^{2}= - 4c

^{2}x

_{1}y

_{1}

Hence locus of (x

_{1}, y

_{1}) is

(x - y)

^{2}+ 4c

^{2}xy = 0

Q.6. Find the co-ordinates of the foci & equation of the directrices of rectangular hyperbola xy = c

^{2}?

Ans: When the hyperbola x

^{2}- y

^{2}= a

^{2}converts into xy = c

^{2}by rotation by - 45

^{0}then a

^{2}= 2c

^{2}.

So, for = 1

Coordinates of foci are (± ae, 0) = (± a, 0)

Also directrices are x = ± = ±

Now we replace for (x, y) & 2c

^{2}= a

^{2}.

Foci = = ± a & = 0

x = y = ± 2a

So Foci are (± 2a, ± 2a)

i.e. (± c, ± c)

Also directrices are

= ±

x + y = ± a

x + y = ± c ( a

^{2}= 2c

^{2})

are the directrices of the rectangular hyperbola.