Elementary Concepts of Hyperbola

Mathematics Class XI Chapter 2: Conic Sections Elementary Concepts of Hyperbola

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Definition:

A hyperbola is defined as the locus of a point moving in a plane in such a way that the ratio of its distance from a fixed point to that from a fixed line is a fixed constant greater than $1$ . i.e ( Eccentricity $(e) > 1$ )

Standard equation of hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ , where $b^2=a^2 (e^2-1)$

$\to$ Eccentricity $(e): e^2 =1 +\dfrac{b^2}{a^2}$

$\to$ Focii $: F (ae, 0)$ and $F'= (-ae ,0)$

$\to$ Equation of directrices : $x=\dfrac{a}{e}$ and $x =-\dfrac{a}{e}$

Transverse Axis:

The line segment $A'A$ of length $2a$ in which the focii $F$ and $F'$ both lie is called the transverse axis of hyperbola.

Conjugate Axis:

The line segment $B'B$ of length $2b$ between the two points $B' =(0, -b)$ and $B = (0, b)$ is called as conjugate axis of the hypoerbola.

Principal Axes:

The transverse and conjugate axis together are called principal axes of the hyperbola.

Vertices: $A =(a, 0)$ and $A'= (-a,0)$

Focal Chord:

A chord that passes through a focus is called a focal chord.

Double Ordinate:

A chord perpendicular to the transverse axis is called double ordinate.

Latus Rectum (L.R):

The focal chord perpendicular to the transverse axis is called the latus rectum. Its length is given by $l= \dfrac{(2b)^2}{a}=\dfrac{(\text {Conjugate Axis})^2}{(\text {Transverse Axis})}= 2a (e^2-1)$

Centre:

The point which bisects every chord of the conic, drawn through it, is called the centre of the conic. $0= (0, 0)$ , the origin is the centre of the hyperbola $\dfrac{x^2}{a^2}- \dfrac{y^2}{b^2}=1$

Note:

(i) Length of L.R $= 2e \times (\text {distance of focus from corresponding directtix})$

(ii) End point of L.R are $: L = \left( ae , \dfrac{b^2}{a} \right), L'= \left( ae, -\dfrac{b^2}{a} \right), M= \left( -ae, \dfrac{b^2}{a} \right), M'= \left( -ae, -\dfrac{b^2}{a} \right)$

Conjugate Hyperbola:

Two hyperbolas are such that the transverse and conjugate axes of the hyperbola are respectively the conjugate and the transverse axes of the other hyperbolas are called conjugate hyperbolas of each other. Example: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ and $-\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ are conjugate hyperbolas.

Equation $: \dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1$

$a^2=b^2(e^2-1) \implies e=\sqrt{1+\dfrac{a^2}{b^2}}$

Vertices: $(0, \pm b); \quad l(\text {L.R})= \dfrac{2a^2}{b}$

Note:

(i) If $e_1$ and $e_2$ are the eccentricities of the hyperbola and its conjugate then $e_1 ^{-2}+e_2^ {-2}=1$

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

(iii) Two hyperbolas are said to be similar if they have the same eccentricity.

(iv) Two similar hyperbolas are said to be equal if they have the same latus rectum.

(v) If a hyperbola is equilateral then the conjugate hyperbola is also equilateral.

$\to$ Since the fundamental equation to hyperbola only differs from that to ellipse in having $-b^2$ instead of $b^2$ , it will be found that many propositions for hyperbola are derived from those for ellipse by simply changing the sign of $b^2$ .

Example:

Find the eccentricity of the hyperbola whose L.R is half of its transverse axis.

Solution:

Equation of hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$

Hence transverse axis $=2a$ and L.R $= \dfrac{2b^2}{a}$

According to question $\dfrac{2b^2}{a}=\dfrac{1}{2}(2a)$

$\implies 2b^2=a^2$

$\implies 2a^2(e^2-1)=a^2 (\therefore b^2=a^2(e^2-1))$

$\implies 2e^2-2=1 \implies e^2=\dfrac{3}{2}$

$\therefore e=\sqrt{\dfrac{3}{2}}$

Parametric Representation of Hyperbola:

The equations $x=a\sec \theta$ and $y=b \tan \theta$ together represent the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ , where $\theta$ is a parameter.

If $P (\theta)=(a \sec \theta , b \tan \theta)$ is on the hyperbola then $Q (\theta)=(a \cos \theta, a \sin \theta)$ is on the auxiliary circle.

The equation to the chord of the hyperbola joining the two-point $P (\alpha)$ and $Q (\beta)$ is given by $\dfrac{x}{a} \cos \dfrac{\alpha -\beta}{2}-\dfrac{y}{b}\sin \dfrac{\alpha+\beta}{2}=\cos \dfrac{\alpha +\beta}{2}$

Auxiliary Circle of Hyperbola:

A circle drawn with centre $O$ and transverse axis diameter is called the auxiliary circle of the hyperbola. Equation of the auxiliary circle is $x^2+y^2=a^2$

Position of the Point w.r.to a Hyperbola:

The quantity $F_1= \dfrac{x^2_1}{a^2}-\dfrac{y_1^2}{b^2}-1$ is positive, zero or negative according as the point $(x_1, y_1)$ lies inside, on or outside the curve.

Example:

Find the position of the point $(5, -4)$ relative to hyperbola $9x^2-y^2=1$ .

Solution:

$9 (5)^2-(-4)^2-1=225-16-1=208 >0$

$\therefore$ So the point $(5, -4)$ lies inside the hyperbola $9x^2-y^2=1$

Rectangular Hyperbola (Equilateral Hyperbola)

The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equally called a rectangular hyperbola/ equilateral hyperbola.

Since $a=b$ equation of hyperbola is $x^2-y^2=a^2$ , whose asymptotes are $y=\pm x$ Eccentricity $(e)=\sqrt{1+\dfrac{b^2}{a^2}}= \sqrt{1+\dfrac{a^2}{a^2}}=\sqrt{2}$

Rotation of this system through an angle of $45 ^o$ in clockwise direction gives another form of the equation of rectangular hyperbola which $xy=c^2$ , where $c^2=\dfrac{a^2}{2}$ .

it is referred to as its asymptotes as axes of co-ordinates.

Vertices: $(c, c)$ and $(-c , -c)$

Focii: $(\sqrt{2}c, \sqrt{2}c)$ and $(-\sqrt{2}c, -\sqrt{2}c)$

Directrices: $x+y=\pm \sqrt {2}c$

Latus Rectum: $: l =2 \sqrt {2}c$ = Transversal Axis = Conjugate Axis

Parametric Equation : $x=ct, y \dfrac{c}{t}, t \in R - \{0\}$

Line and a Hyperbola:

The straight line $y=mx+c$ is a secant, a tangent or passes outside the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ according as $: c^2 > a^2m^2-b^2$ or $c^2 =a^2m^2-b^2$ or $c^2 < a^2m^2-b^2$

Note:

The equation of the chord of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ joining the two points $P (\alpha)$ and $Q (\beta)$ given by $\dfrac{x}{a}\cos \dfrac{\alpha-\beta}{2}-\dfrac{y}{b}\sin \dfrac{\alpha+\beta}{2}=\cos \dfrac{\alpha +\beta}{2}$

Line and a Rectangular Hyperbola:

Equation of a chord joining the points $P (t_1)$ and $Q (t_2)$ is $x+t_1t_2y=c(t_1+t_2)$ .

Equation of the tangent at $P(x_1 , y_1)$ is $\dfrac{x}{x_1}+\dfrac{y}{y_1}=2$ and at $P (t)$ is $\dfrac{x}{t}+ty=2c$

Tangent to Hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$

(i) Slope form:

$y=mx \pm \sqrt{a^2m^2-b^2}$ can be taken as the tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ , having slope $'m'$ .

(ii) Point Form:

Equation of tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ at the point $(x_1 ,y_1)$ is

$\dfrac{xx_1}{a^2}-\dfrac{yy_1}{b^2}=1$

(iii) Parametric Form:

Equation of the tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ at the point $(a \sec \theta, b \tan \theta)$ is $\dfrac{x\sec \theta}{a}-\dfrac{y \tan \theta}{b}=1$

Note:

(i) Point of intersection o the tangents at $P (\theta_1)$ and $Q (\theta _2)$ is $\left( a \dfrac{\cos \dfrac{\theta _1-\theta_2}{2}}{\cos \dfrac{\theta_1+\theta_2}{2}} , b \tan \dfrac{\theta_1+\theta_2}{2}\right)$

(ii) If $|\theta_1+\theta_2|=\pi$ , then tangents at these points $(\theta_1 \text { and } \theta_2)$ are parallel.

(iii) There are two parallel tangents having the same slope $m$ . These tangents touch the hyperbola at extremities of a diameter.

Example:

Find $C$ , if $x+y=c$ touch the hyperbola $\dfrac{x^2}{4}-y^2=1$

Solution:

$x^2-4y^2=4$

$\implies x^2-4(c-x)^2=4$

$\implies 3x^2+8cx +4c^2+4=0$

$D=0$

$64c^2-4 \times 3\times 4(c^2-1)=0$

$\implies c^2-3=0 \implies c = \pm \sqrt{3}$

Normal to Hyperbola:

(i) The equation of the normal to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ at the point $P (x_1 , y_1)$ on it is $\dfrac{a^2x}{x_1}+\dfrac{b^2y}{y_1}=a^2+b^2=a^2e^2$

(ii) The equation of the normal at the point $P (a \sec \theta, b \tan \theta)$ on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is $\dfrac{ax}{\sec \theta}+\dfrac{by}{\tan \theta}=a^2+b^2=a^2e^2$

(iii) Equation of normals in term of their slopes $'m'$ are $y=mx \pm \dfrac{(a^+b^2)m}{\sqrt{a^2-b^2m^2}}$

(iv) Normal to a rectangular hyperbola

Equation of the normal at $P (t)$ is $xt^3-yt= c (t^4-1)$

Some important properties of Hyperbola:

(i) Difference of focal distances is a constant. i.e $2a$

(ii) Locus of the feet of the perpendicular drawn from focus of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ upon any tangents is its auxiliary circle, i.e., $x^2+y^2 =a^2$ and the product of these perpendicular is $b^2$ .

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

(iv) The focii of the hyperbola and the points $P$ and $Q$ in which any tangent at the vertices are concyclic with $PQ$ as diameter of the circle.

Comparative Study of Two Hyperbolas:

	Hyperbola	$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$	$\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1$
1	Centre	$(0, 0)$	$(0, 0)$
2	Foci	$F (ae,0) ,\quad F' (ae,0)$	$F (0 ,be) , \quad F' (0, -be)$
3	Directices	$d=x =\dfrac{a}{e}, \quad d' =x =-\dfrac{a}{e}$	$d=y =\dfrac{b}{e}, \quad d'=y=-\dfrac{b}{e}$
4	Vertices	$A (a, 0), \quad A' (-a ,0)$ $B (0, b), \quad B' (0, -b)$	$B (0, b), \quad B' (0, -b)$ $A (a, 0), \quad A, (-a, 0)$
5	Axes	Transverse axis, $AA' =2a$ Conjugate Axis, $BB' =2b$	Transverse Axis, $BB'=2b$ Conjugate Axis , $AA'=2a$
6	Latus Rectum	$\dfrac{2b^2}{a}$	$\dfrac{2a^2}{b}$
7	Relation between $a , b$ and $e$	$b^2=a^2(e^2-1)$	$a^2=b^2(e^2-1)$

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