DEFINITION

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

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

(i) The eccentricity e is given by the relation $e^2=\left(1+\frac{b^2}{a^2}\right)$

(ii) 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

(iii) The points A and A' are called the vertices of the hyperbola.

(iv) The straight line joining the vertices is called the transverse axis of the hyperbola, its length AA' is 2a.

(v) The straight line BB' is called the conjugate axis.

Illustration 1: Find the eccentricity of the hyperbola which passes through (3, 0) and ($3\sqrt{2},2$).

Solution: Let the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\because$ It passes through (3, 0) and (3$\sqrt{2}$, 2)

$\therefore$ $\frac{9}{a^2}-0=1$ and $\frac{18}{a^2}-\frac{4}{b^2}$ = 1

Which give a² = 9 and b² = 4

$\therefore$ from b² = a²(e² –1), we get 4 = 9(e² –1)

or e² = $\frac{13}{9}$ or e = $\frac{\sqrt{13}}{3}$.

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.

RELATIVE POSITION OF A POINT WITH RESPECT TO THE HYPERBOLA

The quantity $\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$ is positive, zero or negative, according as the point (x₁, y₁) lies within, upon or without the curve.

PARAMETRIC COORDINATES

Any point on the curve, in parametric form is

X = a sec$\theta$, y = b tan$\theta$.

In other words, (a sec$\theta$, b tan $\theta$) is a point on the hyperbola for all values of $\theta$. The point (a sec$\theta$, b tan$\theta$) is briefly called the point '$\theta$'.

KEY 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 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are

(i) The tangent at any point (x₁, y₁) on the curve is $\frac{x{x}_1}{a^2}$ - $\frac{y{y}_1}{b^2}=1$

(ii) The tangent at the point '$\theta$' is $\frac{x\text{ sec}\theta }{a}-\frac{y\text{ tan}\theta }{b}=1$

(iii) The straight line y = mx + c is a tangent to the curve, if c² = a² m² – b². In other words,

y = mx$\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.

(iv) The straight line lx + my = n is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$ = 1 if n² = a²l² – b²m².

(v) Equation of the normal at any point (x₁, y₁) to the curve is $\frac{X-X_1}{X_1/a^2}=\frac{Y-Y_1}{Y_1/\left(-b^2\right)}$

(vi) The equation of the chord through the points $\theta$₁ and $\theta$₂ is

(vii) \left| \begin{gathered} \,\,x  y  1\, \hfill \\ \,\,a{\text{ sec}}{\theta _{\text{1}}} b{\text{ tan}}{\theta _{\text{1}}}\, \, 1\,\, \hfill \\ \,\,a{\text{ sec}}{\theta _{\text{2}}} b{\text{ tan}}{\theta_{\text{2}}} 1 \hfill \\ \end{gathered} \right|\, = 0

(viii) The equation of the normal at $\theta$ is ax cos$\theta$ + by cot$\theta$ = a² + b²

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

(x) Tangent drawn at any point bisects the angle between the lines, joining the point to the foci , whereas normal bisects the supplementary angle between the lines.

(xi) Equation of director 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.

(xii) A straight line x cos a + y sin a = p is tangent if p² = a² cos² $\alpha$ – b² sin² $\alpha$.

(xiii) Equation of straight line passing through the point (a sec$\alpha$, b tana) and (a sec $\theta$', b tan $\theta$') is $\frac{x}{a}\cos \frac{{\theta }^{\prime }-\theta }{2}-\frac{y}{b}\sin \frac{{\theta }^{\prime }+\theta }{2}=\cos \frac{{\theta }^{\prime }+\theta }{2}$.

Illustration 2: Tangents are drawn from point P on the curve x² – 4y² = 4 to the curve x² + 4y² = 4 touching it in the points Q and R. Prove that the mid – point of QR lies on $\frac{x^2}{4}-y^2{\left(\frac{x^2}{4}+y^2\right)}^2$

Solution: Any point on the curve x² – 4y² = 4 can be expressed as ( 2 sec$\theta$, tan$\theta$)

Let (h, k) be the mid – point of QR which is a chord to the curve x² + 4y² = 4

Equation of QR is

T= S₁

$\frac{hx}{4}+ky=\frac{h^2}{4}+k^2$

QR is also the chord of contact to the curve x² + 4y² = 4 w.r.t the point ( 2 sec $\theta$, tan$\theta$)

$\therefore$ Equation of chord of contact of (2 sec $\theta$, tan $\theta$) w.r.t x² + 4y² = 4 is

sec$\theta$ . $\frac{x}{2}$+tan$\theta$ y =1

Comparing the two equation

sec$\theta$ = $\frac{h}{2\left(\frac{h^2}{4}+k^2\right)};\tan \theta =\frac{k^2}{\frac{h^2}{4}+k^2}$

Eliminating sec² $\theta$ – tan² $\theta$ =1

$\frac{h^2}{4{\left(\frac{h^2}{4}+k^2\right)}^2}-\frac{k^2}{{\left(\frac{h^2}{4}+k^2\right)}^2}=1$

$\frac{h^2}{4}-k^2={\left(\frac{h^2}{4}+k^2\right)}^2$

$\therefore$ Required locus of (h, k) is $\frac{x^2}{4}-y^2=\left(\frac{x^2}{4}+y^2\right)$