# Standard Equation of Hyperbola & Relative Position of a Point w.r.t Hyperbola

BASIC CONCEPT

A hyperbola is the locus of a point which moves such that, ratio of its distance from a fixed point (focus) and its distance from a fixed straight line (directrix), is a constant (eccentricity).

This constant (eccentricity) is greater than unity.

#### Standard Equation and Basic Definitions:

Let S be the focus and ZM the directrix of a hyperbola.

Since e > 1, we can divide SZ internally and externally in the ratio e : 1, let the points of division be A and A’ as in the figure.

Let AA’ = 2a and C is the mid point of AA’

Then, SA = e. AZ

and SA’ = e. ZA’

⇒ SA + SA’ = e(AZ + ZA’)

= 2ae

i.e., 2SC = 2ae or SC= ae.

Similarly by subtraction,

SA’ − SA = e(ZA’ − ZA) = 2e(ZC)

⇒ 2a = 2e(ZC)

⇒ ZC = a/e

Now, take C as the origin , CS as the x-axis, and the perpendicular line CY as the y axis.

Then, S is the point (ae , 0) and ZM the line x = a/e.

Let P(x , y) be any point on the hyperbola.

Then the condition PS2 = e2. (distance of P from ZM)2 gives

(x − ae)2 + y2 = e2 (x − a/e)2

x2(1 – e)2 + y2 = a2(1 – e2)

$\displaystyle \frac{x^2}{a^2} – \frac{y^2}{a^2(e^2 -1)} = 1$ …(i)

Since e > 1, e2 – 1 is positive.

Let a2(e2 − 1) = b2 .

Then the equation (i) becomes

$\displaystyle \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$

∎ The eccentricity e of the hyperbola $\displaystyle \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$ is given by the relation

$\displaystyle e^2 = (1+ \frac{b^2}{a^2})$

∎ Since the curve is symmetrical about the y – axis, it is clear that there exists another focus S’ at (−ae, 0) and a corresponding directirx Z’M’ with the equation x= −a/e , such that the same hyperbola is described if a point moves so that its distance from S’ is e times its distance from Z’M’.

∎ The points A and A’ where the straight line joining the two foci cuts the hyperbola are called the vertices of the hyperbola.

∎ The straight line joining the vertices is called the transverse axis of the hyperbola, its length AA’ is 2a

∎ The middle point C of AA’ possesses the property that it bisects every chord of the hyperbola passing through it. It can be proved by taking P(x1 , y1) as any point on hyperbola.

If (x1 , y1) lies on the hyperbola then so does P'(−x1 , −y1) because hyperbola is symmetric about x and y axes. Therefore PP’ is a chord whose middle point is (0, 0), i.e. the origin O. On account of this property the middle point of the straight line joining the vertices of the hyperbola is called the centre of the hyperbola.

∎ The straight line through the centre of a hyperbola which is perpendicular to the transverse axis does not meet the hyperbola in real points. If B and B’ be the points on this line such that BC = CB’ = b , the line BB’ is called the conjugate axis.

∎ The latus rectum is the chord through a focus at right angle to the transverse axis.

∎ The length of the semi-latus rectum obtained by putting x = ae in the equation of the hyperbola is

$\displaystyle y = b\sqrt{\frac{a^2 e^2}{a^2} – 1}$

$\displaystyle y = b\sqrt{e^2 – 1} = \frac{b^2}{a}$

###### Relation Between Focal Distances

The difference of the focal distances of a point on the hyperbola is constant. PM and PM’ are perpendiculars to the directrices MZ and M’Z’

PS’ − PS = e(PM’ − PM)

= eMM’ = e(2a/e)

= 2a = constant.

Remark:

∎ Using this property hyperbola can be defined in another way − Locus of a moving point such that difference of its distances from two fixed point is constant, would be hyperbola.

For example, interference fringes formed in Young’s Double Slit experiment are hyperbolic in nature.

###### Relative Position of a Point with respect to the Hyperbola:

The quantity $\displaystyle \frac{x_1^2}{a^2} – \frac{y_1^2}{b^2} = 1$ is positive, zero or negative

i.e. ( S1   >  = or < 0 ) , according as the point (x1, y1) lies within, upon or without the curve, where

S1 = $\displaystyle S_1 = \frac{x_1^2}{a^2} – \frac{y_1^2}{b^2} – 1$

Note that this relation is converse in hyperbola if we compare with that in circle, parabola or ellipse.

### Parametric Coordinates:

We can express the coordinates of a point of the hyperbola $\displaystyle \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$
in terms of a single parameter, say θ
In the adjacent figure OM = a secθ and PM = b tanθ.

Thus any point on the curve , in parametric form is x = a secθ, y = b tanθ

In other words, (a secθ, b tanθ) is a point on the hyperbola for all values of θ. The point (a secθ, b tanθ) is briefly written the point ‘θ’

##### Important Properties of Hyperbola:

Since the fundamental equation of the hyperbola only differs from that of the ellipse in having −b2 instead of b2 , it will be found that many propositions for the hyperbola are derived from those for the ellipse by changing the sign of b2 .

Some results for the hyperbola $\displaystyle \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$
are :

(i) The tangent at any point (x1, y1) on the curve is $\displaystyle \frac{x x_1}{a^2} – \frac{y y_1}{b^2} = 1$

(ii) The tangent at the point ‘ θ ‘ is $\displaystyle \frac{x sec\theta}{a} – \frac{y tan\theta}{b } = 1$

(iii) The straight line y = mx + c is a tangent to the curve , if c2 = a2 m2 − b2 .
In other words , $\displaystyle y = m x \pm \sqrt{a^2 m^2 – b^2}$
touches the curve, for all those values of m when m > b/a or m < − b/a.

If m = ±b/a the tangents touch at infinity and are called as asymptotes.

(iv) Equation of the normal at any point (x1, y1) to the curve is $\displaystyle \frac{x-x_1}{x_1^2 /a^2} = \frac{y-y_1}{y_1/(-b^2)}$

(v) The equation of the chord through the points θ1 and θ2 is

$\large \left| \begin{array}{ccc} x & y & 1 \\ a sec\theta_1 & b tan\theta_1 & 1 \\ a sec\theta_2 & b tan\theta_2 & 1\end{array} \right| =0$

(vi) The equation of the normal at θ is ax cosθ + by cotθ = a2 + b2

(vii) Through a given point, four normals can be drawn to a hyperbola (real or imaginary).

(viii) Normal drawn at any point bisects the angle between the lines, joining the point to the foci, whereas the tangent at the same point bisects the supplementary angle between the lines.

(ix) Equation of director the circle is x2 + y2 = a2 – b2 .
That means if a2 > b2, there would exist several points such that tangents drawn from them would be mutually perpendicular.

If a2 < b2 , no such point exist.

For a2 = b2 , centre is the only point from which two perpendicular tangents (asymptotes) to the hyperbola can be drawn.