Hyperbola Notes for IIT JEE, Download PDF!

Hyperbola is an important topic from JEE point of view. Conic section constitutes 3-4 questions every year in JEE Main in which one question is from hyperbola. Read and revise all the important topics from hyperbola. Download the pdf of the Short Notes on Hyperbola from the link given at the end of the article.

1. Hyperbola

Hyperbola is the locus of a point in a plane such that the difference of its distance from two fixed point in the same plane is always constant.

Here, e> 1 and h² >ab

Let P(h,k)

Thus, according to the definition, equation would be

|PF₁ | - |PF₂| = 2a

Solving this condition will give the equation of hyperbola as:

, where b² = a² (e² – 1)

2. Nomenclature of hyperbola:

Eccentricity (e):

Foci: (c,0) and (-c,0) or (ae, 0) and (-ae,0)

Transverse axis (TA): The line segment containing the foci is known as the transverse axis and the length is 2a

Conjugate axis (CA): The line segment containing the points B₁ and B₂ is called as conjugate axis and the length is 2b.

Center of the hyperbola: Point of the intersection of TA and CA is known as the center.

Both TA and CA are together known as the principal axes of the hyperbola.

Vertices: A₁ (a,0) and A₂ (-a,0)

Focal axis: Line containing the fix points F₁ and F₂ (called as foci) is called as the focal axis

Focal length: The distance between F₁ and F₂ is called the focal length

Focal chord: A chord passing through a focus is called the focal chord.

Latus rectum (LR): LR is the focal chord which is perpendicular to TA and whose length is 2b²/a = 2a(e² – 1)= 2e ( ae – a/e) = 2e (Distance between focus and corresponding foot of directrix)

Double ordinate: A chord which is perpendicular to the transverse axis.

Directrix: x = ±a/e

3. Conjugate Hyperbola

Corresponding to every hyperbola there exist a hyperbola such that the conjugate axis and the transverse axis of one is equal to the transverse axis and the conjugate axis of the other. Such hyperbolas are called as the conjugate hyperbola

If, hyperbola, H:

Then conjugate hyperbola, CH:

If e₁ is the eccentricity of the hyperbola and e₂ is the eccentricity of the conjugate hyperbola then

, whereas e₁² = 1 + and e₁² = 1 +

4. Rectangular Hyperbola

A hyperbola will be converted to a rectangular hyperbola if a = b i.e., TA = CA. This hyperbola is also known as an equilateral hyperbola.

Eccentricity, e =√2

Length of latus rectum= 2a = CA = TA

5. Auxiliary Circle and Eccentric angle

The circle described on the transverse axis of the hyperbola as the diameter is called an auxiliary circle.

Thus it’s equation would be

x² + y² = a²  

And for conjugate hyperbola, the equation would be

x² + y² = b²

6. Parametric coordinates of a point on the hyperbola

P (a sec θ,  b tan θ)

From point P, a perpendicular line is drawn to meet the TA at  (let’s say).

From 'T' a tangent is drawn on the auxiliary circle, meeting the circle at Q.

These two points: P and Q are called as the corresponding point and Q is called the eccentric angle of the point P.

7. The position of a point with respect to the hyperbola

Point P lies outside the hyperbola,

S₁ < 0 : Point is outside the curve

S₁ > 0: Point is inside the curve

S₁ = 0: Point is on the curve

8. Line and Parabola

Let, L: y = mx + c

S:

A line can be a tangent to the hyperbola, it may cut the hyperbola or it may not cut/touch the hyperbola at all.

To find that, we will make a quadratic in ‘x’ using equations of L and S.

If discriminant D> 0 ⇒ Two roots ⇒ Intersect ⇒ Secant

If D= 0 ⇒ One root ⇒ Touching the ellipse⇒ Tangent

If D< 0 ⇒ No roots ⇒ Neither secant nor tangent

9. Condition of Tangency

D = 0

⇒

Thus the equation of the tangent would be

L: y = mx + c

10. Number of tangents from a given point (h,k) to the hyperbola

We have,  

P (h,k)

Thus,

(k – mh)² =a²m² - b²  which is a quadratic in m.

This suggests that from a given point P(h,k) we can draw at maximum two tangents.

11. The angle between the two tangents

Let the angle between the tangents be θ

 From quadratic equation, we know slope: m₁ and m₂

Thus,

From the quadratic equation in m,

12. Director circle of the ellipse

When θ = 90° then m₁m₂ = -1

x² + y² = a² - b² : Equation of director circle.

Now, director circle can be imaginary or real.

Real director circle: when the length of TA > length of CA

Imaginary director circle: when the length of TA < length of CA

Point circle: when the length of TA = length of CA ie., in case of a rectangular hyperbola.  

Thus, director circle is the locus of all those points from where the ellipse can be seen at angle 90°.

13. The equation of the chord of a hyperbola joining α and β on it

14. The equation of Tangent, Normal and Chord of Contact

Tangents:

a) Cartesian Tangent

b) Slope form

c) Parametric form

 Normal:

a)Cartesian Normal

b) Slope form

c) Parametric form

15. Chord of contact

The equation of chord of contact will be similar to that of the tangent. Thus a line when touches the hyperbola will be tangent and the same line when cuts the hyperbola will be the chord of contact.

Equation: T = 0 (Similar to that of tangent equation)

16. Pair of tangents

The equation of pair of tangents would be

SS₁ = T², where S is the equation of the hyperbola, S₁ is the equation when a point P (h,k) satisfies S, T is the equation of the tangent.

17. The equation of the chord whose middle point is (x₁, y₁)

T = S₁

18. Rectangular Hyperbola

This is a special kind of hyperbola when the length of traverse and conjugate axis are equal.

The equation is

x² – y² = a²

However, rotating the coordinate axis through an angle of 45⁰, we get another form of this rectangular hyperbola.

i.e., XY = c²

18.1 Parametric coordinates of (RH)

(ct, c/t ) where t≠ 0 and t ∈ R

Two points on this hyperbola can be represented by (ct₁,c/t₁) and

(ct₂, c/t₂)

18.2 The equation of chord joining t₁ and t₂ points on the rectangular hyperbola

Two points on this hyperbola can be represented by (ct₁,) and (ct₂,)

The slope of the line joining t₁ and t₂ :

Thus the equation of chord:

x + t₁t₂ y = c (t₁ + t₂)

18.3 The equation of tangent of the rectangular hyperbola

a)Cartesian form at the point (x₁,y₁)

b) Parametric form

18.4 The equation of a normal of the rectangular hyperbola

Parametric form:

(y -c/t ) = t² ( x – ct)

 The equation of a chord whose middle point is (h,k)

T = S₁

Thus, the equation is  

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