Use these Inter 2nd Year Maths 2B Formulas PDF Chapter 5 Hyperbola to solve questions creatively.

## Intermediate 2nd Year Maths 2B Hyperbola Formulas

Definition:

A conic with an eccentricity greater than one is called a hyperbola, i.e., the locus of a point, the ratio of whose distances from a fixed point (focus) and fixed straight line (directrix) is e, where e> 1 is called a hyperbola.

Equation of hyperbola in standard form:

→ \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1

Here b^{2} = a^{2} (e^{2} – 1)

→ The difference of the focal distances of any point on the hyperbola is constant.

(i.e.,) SP ~ S’P = 2a

→ A Point P(x_{1}, y_{1}) in the plane of the hyperbola S = 0 lies inside the hyperbola, if S_{11} < 0, lies outside if S_{11} > 0 and on the curve S_{11} = 0

→ Ends of the latus rectum (± ae, ± b^{2}/a) and the length of the latus rectum is 2b^{2}/a.

→ Equation of tangent at P(x_{1}, y_{1}) to \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1

\(\frac{x x_{1}}{a^{2}}-\frac{y y_{1}}{b^{2}}\) = 1

→ Equation of normal at P(x_{1}, y_{1})

\(\frac{a^{2} x}{x_{1}}+\frac{b^{2} y}{y_{1}}\) = a^{2} + b^{2}

→ Parametric form is x = a sec θ, y = b tan θ.

→ Equation of tangent at ‘θ’ on the hyperbola

\(\frac{x}{a}\) sec θ – \(\frac{y}{b}\) tan θ = 1

→ Equation of normal at θ on the hyperbola

\(\frac{a x}{\sec \theta}+\frac{b y}{\tan \theta}\) = a^{2} + b^{2}

→ A necessary and sufficient condition for a straight line y = mx + c to be tangent to hyperbola c^{2} = a^{2}m^{2} – b^{2}

y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)

→ If P(x_{1}, y_{1}) is any point in the plane of the hyperbola S = 0 then the equation of the polar of P is S_{1} = 0.

→ The pole of the line lx + my + n = 0; n ≠ 0 with respect to the hyperbola S = 0 is

\(\left(\frac{-a^{2} l}{n} ; \frac{b^{2} m}{n}\right)\), n ≠ 0

**Equation of a Hyperbola in Standard From.**

The equation of a hyperbola in the standard form is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

Proof:

Let S be the focus, e be the eccentricity and L = 0 be the directrix of the hyperbola.

Let P be a point on the hyperbola. Let M, Z be the projections of P, S on the directrix L = 0 respectively. Let N be the projection of P on SZ. Since e > 1, we can divide SZ both internally and externally in the ratio e : 1.

Let A, A’ be the points of division of SZ in the ratio e : 1 internally and externally respectively. Let AA’ = 2a. Let C be the midpoint of AA’. The points A, A’ lie on the hyperbola and \(\frac{\mathrm{SA}}{\mathrm{AZ}}\) = e, \(\frac{\mathrm{SA}^{\prime}}{\mathrm{A}^{\prime} \mathrm{Z}}\) = e

∴ SA = eAZ, SA’ = eA’Z

Now SA + SA’ = eAZ + eA’Z

⇒ CS – CA + CS + CA’ = e(AZ + A’Z)

⇒ 2CS = eAA’ (∵ CA = CA’)

⇒ 2CS = e2a ^ CS = ae

Aslo SA’ – SA = eA’Z – eAZ

AA’ = e(A’Z – AZ)

⇒ 2a = e[CA’ + CZ – (CA – CZ)]

⇒ 2a = e 2CZ (∵ CA = CA’) ⇒ CZ = \(\frac{a}{e}\).

Take CS, the principal axis of the hyperbola as

x-axis and Cy perpendicular to CS as y-axis. Then S = (ae, 0).

Let P(x_{1}, y_{1}).

Now PM = NZ = CN – CZ = x_{1} – \(\frac{a}{e}\).

P lies on the hyperbola ⇒ \(\frac{\mathrm{PS}}{\mathrm{PM}}\) = e

⇒ PS = ePM ⇒ PS^{2} = e^{2}PM^{2}

⇒ (x_{1} – ae)^{2} + (y_{1} – 0)^{2} = e^{2} \(\left(x_{1}-\frac{a}{e}\right)^{2}\)

⇒ (x_{1} – ae)^{2} + y_{1}^{2} = (x_{1}e – a)^{2}

⇒ x_{1}^{2} + a^{2} e^{2} – 2x_{1}ae + y_{1}^{2} = x_{1}e^{2} + a^{2} – 2x_{1}ae

⇒ x^{2}(e^{2} -1) – y^{2} = a^{2}(e^{2} -1)

⇒ \(\frac{x_{1}^{2}}{a^{2}}-\frac{y_{1}^{2}}{a^{2}\left(e^{2}-1\right)}\) = 1 ⇒ \(\frac{x_{1}^{2}}{a^{2}}-\frac{y_{1}^{2}}{b^{2}}\) = 1

Where b^{2} = a^{2}(e^{2} – 1)

The locus of P is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

The equation of the hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

Nature of the curve \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1

Let C be the curve represented by \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1. Then

(i) (x,y) ∈ C(x, -y) ∈ C and (x, y) ∈ C ⇔ (-x,y) ∈ C.

Thus the curve is symmetric with respect to both the x-axis and the y-axis. Hence the coordinate axes are two axes of the hyperbola.

(ii) (x, y) ∈ C ⇔ (-x, -y) ∈ C.

Thus the curve is symmetric about the origin O and hence O is the midpoint of every chord of the hyperbola through O. Therefore the origin is the center of the hyperbola.

(iii) (x, y) ∈ C and y = 0 ⇒ x^{2} = a^{2} ⇒ x = ±a.

Thus the curve meets x-axis (Principal axis) at two points A(a, 0), A'(-a, 0). Hence hyperbola has two vertices. The axis AA’ is called transverse axis. The length of transverse axis is AA’ = 2a.

(iv) (x, y) ∈ C and x = 0 ⇒ y^{2} = -b^{2} ⇒ y is imaginary.

Thus the curve does not meet the y-axis. The points B(0, b), B'(0, -b) are two points on y-axis. The axis BB’ is called conjugate axis. BB’ = 2b is called the length of conjugate axis.

(v) \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 ⇒ y = \(\frac{b}{a} \sqrt{x^{2}-a^{2}}\) ⇒ y has no real value for -a < x < a.

Thus the curve does not lie between x = -a and x = a.

Further x → ∞ ⇒ y ^ ± ∞ and

x → -∞ ⇒ y → ± ∞.

Thus the curve is not bounded (closed) on both the sides of the axes.

(vi) The focus of the hyperbola is S(ae, 0). The image of S with respect to the conjugate axis is S'(-ae, 0). The point S’ is called second focus of the hyperbola.

(vii) The directrix of the hyperbola is x = a/e. The image of x = a/e with respect to the conjugate axis is x = -a/e. The line x = -a/e is called second directrix of the hyperbola corresponding to the second focus S’.

Theorem:

The length of the latus rectum of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{2 b^{2}}{a}\)

Proof:

Let LL be the length of the latus rectum of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

If SL = 1, then L = (ae, 1)

L lies on the hyperbola ⇒ \(\frac{(a \mathrm{e})^{2}}{a^{2}}-\frac{1^{2}}{b^{2}}\) = 1

⇒ \(\frac{1^{2}}{\mathrm{~b}^{2}}\) = e^{2} – 1 ⇒ 1^{2} = b^{2}(e^{2} – 1)

⇒ 1 = b^{2} × \(\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}\) ⇒ 1 = \(\frac{b^{2}}{a}\) ⇒ SL = \(\frac{b^{2}}{a}\)

∴ LL’ = 2SL = \(\frac{2 b^{2}}{a}\)

Theorem:

The difference of the focal distances of any point on the hyperbola is constant f P is appoint on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 with foci S and S then |PS’- PS| = 2a

Proof:

Let e be the eccentricity and L = 0, L’ = 0 be the directrices of the hyperbola.

Let C be the centre and A, A’ be the vertices of the hyperbola.

∴ AA’ = 2a

Foci of the hyperbola are S(ae, 0), S'(-ae, 0).

Let P(x_{1}, y_{1}) be a point on the hyperbola.

Let M, M’ be the projections of P on the directrices L = 0, L’ = 0 respectively.

∴ \(\frac{\mathrm{SP}}{\mathrm{PM}}\) = e, \(\frac{\mathrm{S}^{\prime} \mathrm{P}}{\mathrm{PM}^{\prime}}\) = e

Let Z, Z’ be the points of intersection of transverse axis with directrices.

∴ MM’ = ZZ’ = CZ + CZ’ = 2a/e

PS’ – PS = ePM’ – ePM = e(PM’ – PM)

= e(MM’) = e(2a/e) = 2a

Notation: We use the following notation in this chapter.

S ≡ \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) – 1, S_{1} = \(\frac{x_{1}}{a^{2}}-\frac{y_{1}}{b^{2}}\) – 1

S_{11} ≡ S(x1, y1) = \(\frac{x_{1}^{2}}{a^{2}}-\frac{y_{1}^{2}}{b^{2}}\) – 1, S_{12} = \(\frac{\mathrm{x}_{1} \mathrm{x}_{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}_{1} \mathrm{y}_{2}}{\mathrm{~b}^{2}}\) – 1

Note: Let F(x_{1,} y_{1}) be a point and S ≡ \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) – 1 = 0 be a hyperbola. Then

- F lies on the hyperbola S = 0 ⇔ S
_{11}= 0 - F lies inside the hyperbola S = 0 ⇔ S
_{11}> 0 - F lies outside the hyperbola S = 0 ⇔ S
_{11}< 0

Theorem:

The equation of the chord joining the two points A(x_{1}, y_{1}), B(x_{2}, y_{2}) on the hyperbola S = 0 is S_{1} + S_{2} = S_{12}.

Theorem:

The equation of the normal to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 at P(x_{1} y_{1}) is \(\frac{\mathrm{a}^{2} \mathrm{x}}{\mathrm{x}_{1}}+\frac{\mathrm{b}^{2} \mathrm{y}}{\mathrm{y}_{1}}\) = a^{2} + b^{2}.

Theorem: The condition that the line y = mx + c may be a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is c^{2} = a^{2}m^{2} – b^{2}

Note: The equation of the tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 may be taken as y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\)

The point of contact is \(\left(\frac{-a^{2} m}{c}, \frac{-b^{2}}{c}\right)\) where c = am – b

Theorem: Two tangents can be drawn to a hyperbola from an external point.

Note: If m_{1}, m_{2} are the slopes of the tangents through P, then m_{1} m_{2} become the roots of (x_{2}^{2} – a^{2})m^{2} – 2x_{1}y_{1}m + (y_{1}^{2} + b^{2}) = 0

Hence m_{1} + m_{2} = \(\frac{2 \mathrm{x}_{1} \mathrm{y}_{1}}{\mathrm{x}_{1}^{2}-\mathrm{a}^{2}}\)

m_{1}m_{2} = \(\frac{\mathrm{y}_{1}^{2}+\mathrm{b}^{2}}{\mathrm{x}_{1}^{2}-\mathrm{a}^{2}}\)

Theorem:

The point of intersection of two perpendicular tangents to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 lies on the circle x^{2} + y^{2} = a^{2} – b^{2}.

Proof:

Equation of any tangent to the hyperbola is:

y = mx ± \(\)

Suppose P(x_{1}, y_{1}) is the point of intersection of tangents.

Plies on the tangent Y_{1} = mx_{1} ± \(\sqrt{a^{2} m^{2}-b^{2}}\) y_{1} = mx_{1} = ±\(\sqrt{a^{2} m^{2}-b^{2}}\)

= (y_{1} – mx_{1})^{2} = a^{2}m^{2} – b^{2}

y_{1}^{2} + m^{2}x_{1}^{2} – 2mx_{1}y_{1} – a^{2}m^{2} + b^{2} = 0

= m^{2}(x_{1}^{2 }– a^{2}) – 2mx_{1}y_{1} + (y_{1}^{2 }+ b^{2}) = 0

This is a quadratic in m giving the values for m say m1 and m2.

The tangents are perpendicular

⇒ m_{1}m_{2} = -1 ⇒ \(\frac{\mathrm{y}_{1}^{2}+\mathrm{b}^{2}}{\mathrm{x}_{1}^{2}-\mathrm{a}^{2}}\) = -1

⇒ y_{1}^{2} + b^{2} = -x^{2} + a^{2} ⇒ x_{1}^{2} + y_{1}^{2} = a^{2} – b^{2}

P(x_{1}, y_{1}) lies on the circle x^{2} + y = a^{2} – b^{2}.

Definition:

The point of intersection of perpendicular tangents to a hyperbola lies on a circle, concentric with the hyperbola. This circle is called director circle of the hyperbola.

**Corollary:**

The equation to the auxiliary circle of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\)= 1 is x^{2} + y^{2} = a^{2}.

Theorem:

The equation to the chord of contact of P(x_{1}, y_{1}) with respect to the hyperbola S = 0 is S_{1} = 0.

**Midpoint of a Chord:**

Theorem: The equation of the chord of the hyperbola S = 0 having P(x_{1}, y_{1}) as it’s midpoint is S_{1} = S_{11}.

**Pair of Tangents:**

Theorem: The equation to the pair of tangents to the hyperbola S = 0 from P(x_{1}, y_{1}) is S_{1}^{2} = S_{11}S

**Asymptotes:**

Definition: The tangents of a hyperbola which touch the hyperbola at infinity are called asymptotes of the hyperbola.

Note:

- The equation to the pair of asymptotes of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 0.
- The equation to the pair of asymptotes and the hyperbola differ by a constant.
- Asymptotes of a hyperbola passes through the centre of the hyperbola.
- Asymptotes are equally inclined to the axes of the hyperbola.
- Any straight line parallel to an asymptote of a hyperbola intersects the hyperbola at only one point.

Theorem:

The angle between the asymptotes of the hyperbola S = 0 is 2tan^{-1}(b/a).

Proof:

The equations to the asymptotes are y = ± \(\frac{b}{a}\)x.

If θ is an angle between the asymptotes, then

tan θ = \(\frac{\frac{\mathrm{b}}{\mathrm{a}}-\left(-\frac{\mathrm{b}}{\mathrm{a}}\right)}{1+\left(\frac{\mathrm{b}}{\mathrm{a}}\right)\left(-\frac{\mathrm{b}}{\mathrm{a}}\right)}=\frac{2\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}\) = tan 2α Where tan α = \(\frac{b}{a}\)

∴ θ = 2α = 2Tan^{-1}\(\frac{b}{a}\)

**Parametric Equations:**

A point (x, y) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 represented as x = a sec θ, y = b tan θ in a single parameter θ. These equations x = a sec0, y= b tan θ are called parametric equations of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

The point (a sec θ, b tan θ) is simply denoted by θ.

Note: A point on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 can also be represented by (a cosh θ, b sinh θ). The equations x = a cosh θ, y = sinh θ are also called parametric equations of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1.

Theorem: The equation of the chord joining two points α and β on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is:

\(\frac{x}{a}\)cos\(\frac{\alpha-\beta}{2}\) – \(\frac{y}{b}\)sin\(\frac{\alpha+\beta}{2}\) = cos\(\frac{\alpha+\beta}{2}\)

Theorem:

The equation of the tangent at P(0) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{x}{a}\)cos\(sec θ – [latex]\frac{y}{b}\) tan θ = 1.

Theorem:

The equation of the normal at P(0) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{a x}{\sec \theta}+\frac{\text { by }}{\tan \theta}\) = a^{2} + b^{2}.