Click to Chat

1800-2000-838

+91-120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: Rs. 15,900
• View Details

```Revision Notes on Hyperbola

The standard equation of hyperbola with reference to its principal axis along the coordinate axis is given by x2/a2 - y2/b2 = 1, where b2 = a2 (e2 -1).

The foci of the hyperbola are S(ae, 0) and S’ = (-ae, 0).

Equations of the directrices are given by x = a/e and x = -a/e.

The coordinates of vertices are A’ = (-a, 0) and A = (a, 0).

The length of latus rectum is 2b2/a = 2a(e2 - 1).

The length of the transverse axis of the hyperbola is 2a.

The difference of the focal distances of any point on the hyperbola is constant and is equal to transverse axis.

x2/a2 - y2/b2 = 1 and -x2/a2 + y2/b2 = 1 are conjugate hyperbola of each other.

If e1 and e2 are the eccentricities of the hyperbola and its conjugate then e1-2 + e2-2 = 1

The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

The length of the transverse axis of a hyperbola is 2a and the transverse axis and conjugate axis together constitute the principal axis of the hyperbola.

Whether two hyperbolas are similar or not is decided on the basis of their eccentricity. The hyperbolas with same eccentricity are same.

The equation of the auxiliary circle of the hyperbola is given by

x2+ y2 = a2

In parametric form, the equations x = a sec θ and y = b tan θ together represent the hyperbola x2/a2 - y2/b2 = 1. Here θ is a parameter.

The point P(x1, y1) lies within, on or outside the ellipse according as

x12/a2 - y12/b2 = 1 is positive, zero or negative.

The line y = mx + c is a secant, a tangent or passes outside the hyperbola  x2/a2 - y2/b2 = 1 according as whether c2 is > = or < a2m2 - b2

?Equation of tangent:

Equation of tangent to hyperbola x2/a2 - y2/b2 = 1at the point (x1, y1) is xx1/a2 - yy1/b2 = 1

Equation of tangent to hyperbola x2/a2 - y2/b2 = 1 at the point (a sec θ, b tan θ) is (x sec θ)/a - (y tan θ)/b = 1

y = mx ± √[(a2m2 - b2] can also be taken as the tangent to the hyperbola x2/a2 - y2/b2 = 1

Equation of normal:

Equation of normal to the hyperbola x2/a2 - y2/b2 = 1 at the point (x1, y1) is a2 x/x1 + b2 y/y1 = a2 + b2 = a2e2

Equation of normal at the point P(a sec θ, b tan θ) on the hyperbola x2/a2 - y2/b2 = 1 is ax/sec θ + by/ tan θ = a2 + b2 = a2e2

?The combined equation of pair of tangents drawn from a point P(x1, y1) lying outside the hyperbola x2/a2 – y2/b2 = 1 is SS1 = T2, where S = x2/a2 – y2/b2 - 1, x12/a2 – y12/b2 - 1 and T = xx1/a2 – yy1/b2 - 1

The tangent and normal at any point of a hyperbola bisect the angle between the focal radii.

The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.

The equation of director circle is x2 + y2 = a2 - b2

The equations of the asymptotes of the hyperbola are
x/a + y/b = 0 and x/a - y/b = 0.
This can be combined as x2/a2 - y2/b2 = 0

The asymptotes of the hyperbola and its conjugate are same.

Comparison between hyperbola and its conjugate hyperbola:?

Basic Elements

Hyperbola

Conjugate Hyperbola

x2/a2 - y2/b2 = 1

x2/a2 - y2/b2 = -1

Centre

(0,0)

(0,0)

Length of transverse axis

2a

2b

Length of conjugate axis

2b

2a

eccentricity

b2= a2(e2-1)

a2= b2(e2-1)

Foci

(± ae,0)

(0, ± be)

Equation of directrix

x = ± a/e

y = ± b/e

Length of latus rectum

2b2/a

2a2/b

Difference of focal distances

2a

2b

Equation of transverse axis

y = 0

x = 0

equation of conjugates

x = 0

y = 0

Tangent at vertices

x = ± a

y = ± b

The asymptotes pass through the center of the hyperbola and the bisectors of the angles between the asymptotes are the axis of a hyperbola

If the lengths of transverse and conjugate axis are equal, then such a hyperbola is termed to be a rectangular hyperbola.

The eccentricity of rectangular hyperbola is √2 and the length of its latus rectum is equal to its transverse or conjugate axis.

The equation xy = c2 represents a rectangular hyperbola

In a hyperbola b2 = a2 (e2 – 1). In the case of rectangular hyperbola (i.e., when b = a) result become a2 = a2(e2 – 1) or e2 = 2 or e =√2

In parametric form, its coordinates are x = ct, y = c/t, t ∈ R ~ {0}

Equation of a chord joining the points P(t1) and Q(t2) is x + t1t2y = c(t1 + t2) with slope m = -1/ t1t2

Equation of tangent at P (x1, y1) is x/x1 + y/y1 = 2

Equation of tangent at P(t) is x/t + ty = 2c

Equation of normal is y - c/t = t2(x – ct)

Chord whose middle point is given to be (h, k) is kx + hy = 2hk

The equation of the director circle of the hyperbola is given by x2 + y2 = a2 - b2.

If b2 < a2, this circle is real.

If b2 = a2 the radius of the circle is zero and it reduces to a point circle at the origin. In this case, centre is the only point from which the tangents at right angles can be drawn to the curve.

If b2 > a2 the radius of the circle is imaginary, so that there is no such circle and so no tangents at right angle can be drawn to the curve.

```