HYPERBOLA
The Hyperbola is a conic whose eccentricity is greater than unity \((e>1) .\)
1. STANDARD EQUATION & DEFINITION(S):
where \(b^{2}=a^{2}\left(e^{2}-1\right)\)
or \(a^{2} e^{2}=a^{2}+b^{2}\) i.e. \(e^{2}=1+\dfrac{b^{2}}{a^{2}}\)\(=1+\left(\dfrac{\text { Conjugate Axis }}{\text { Transverse Axis }}\right)^{2}\)
(a) Foci :
\(\mathrm{S} \equiv(\mathrm{a} e, 0) \quad \& \quad \mathrm{~S}^{\prime} \equiv(-\mathrm{a} e, 0) .\)
(b) Equations of directrices:
\(\mathrm{x}=\dfrac{\mathrm{a}}{e}\quad \) & \(\quad \mathrm{x}=-\dfrac{\mathrm{a}}{e}\)
(c) Vertices:
\(A \equiv(a, 0)\quad \) & \(\quad A^{\prime} \equiv(-a, 0)\)
(d) Latus rectum:
(i) Equation: \(\mathrm{x}=\pm \mathrm{ae}\)
(ii) Length: \(\begin{aligned} &=\dfrac{2 b^{2}}{a}=\dfrac{(\text { Conjugate Axis })^{2}}{(\text { Transverse Axis })}=2 \text { a }\left(e^{2}-1\right) \\ &=2 e \text { (distance from focus to directrix) } \end{aligned}\)
(iii) Ends : \(\left(a e, \dfrac{b^{2}}{a}\right),\left(a e, \dfrac{-b^{2}}{a}\right) ;\left(-a e, \dfrac{b^{2}}{a}\right),\left(-a e, \dfrac{-b^{2}}{a}\right)\)
(e) (i) Transverse Axis:
The line segment \(A^{\prime} A\) of length \(2 a\) in which the foci \(S^{\prime} \) & \(S\) both lie is called the Transverse Axis of the Hyperbola.
(ii) Conjugate Axis:
The Transverse Axis & the Conjugate Axis of the hyperbola are together called the Principal axis of the hyperbola.
(f) Focal Property:
The difference of the focal distances of any point on the hyperbola is constant and equal to transverse axis i.e. || \(\mathrm{P} \mathrm{S}|-| \mathrm{PS}^{\prime}||=2 \mathrm{a}\). The distance \(\mathrm{SS}^{\prime}=\) focal length.
(g) Focal distance :
Distance of any point \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) on hyperbola from foci \(\mathrm{PS}=\mathrm{ex}-\mathrm{a}\) & \( \mathrm{PS}^{\prime}= \mathrm{ex}+\mathrm{a}\)
2. CONJUGATE HYPERBOLA:
Two hyperbolas such that transverse & conjugate axis of one hyperbola are respectively the conjugate & the transverse axis of the other are called Conjugate Hyperbolas of each other. eg. \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) & \(-\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\) are conjugate hyperbolas of each other.
(i) If \(e_{1} \) & \(e_{2}\) are the eccentricities of the hyperbola & 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.
3. RECTANGULAR OR EQUILATERAL HYPERBOLA:
The particular kind of hyperbola in which the lengths of the transverse & conjugate axis are equal is called an Equilateral Hyperbola.
Note that the eccentricity of the rectangular hyperbola is \(\sqrt{2}\) and the length of it's latus rectum is equal to it's transverse or conjugate axis.
4. AUXILIARY CIRCLE :
Note from the figure that \(\mathrm{P} \) & \( \mathrm{Q}\) are called the "Corresponding Points" on the hyperbola & the auxiliary circle. '\(\theta\)' is called the eccentric angle of the point 'P' on the hyperbola. \((0 \leq \theta<2 \pi)\).
Parametric Equation:
The equations \(\mathrm{x}=\mathrm{a} \sec \theta \) & \(\mathrm{y}=\mathrm{b} \tan \theta\) together represents the hyperbola \(\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\) where \(\theta\) is a parameter.
5. POSITION OF A POINT 'P' w.r.t. A HYPERBOLA:
The quantity \(\dfrac{\mathrm{x}_{1}{ }^{2}}{\mathrm{a}^{2}}-\dfrac{y_{1}{ }^{2}}{\mathrm{~b}^{2}}=1\) is positive, zero or negative according as the point \(\left(x_{1}, y_{1}\right)\) lies within , upon or outside the curve.
6. LINE AND A HYPERBOLA :
The straight line \(y=m x+c\) is a secant, a tangent or passes outside the hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) according as: \( c^{2}>=<a^{2} m^{2}-b^{2}\).
Equation of a chord of the hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) joining its two points \(\mathrm{P}(\alpha) \) & \( \mathrm{Q}(\beta)\) is \(\dfrac{\mathrm{x}}{\mathrm{a}} \cos \dfrac{\alpha-\beta}{2}-\dfrac{\mathrm{y}}{\mathrm{b}} \sin \dfrac{\alpha+\beta}{2}=\cos \dfrac{\alpha+\beta}{2}\)
7. TANGENT TO THE HYPERBOLA \(\mathbf{\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}\) :
Note : In general two tangents can be drawn from an external point \(\left(x_{1} ,y_{1}\right)\) to the hyperbola and they are \(y-y_{1}=m_{1}\left(x-x_{1}\right) \) & \(y-y_{1}=m_{2}\left(x-x_{2}\right)\), where \(m_{1} \) & \(m_{2}\) are roots of the equation \(\left(\mathrm{x}_{1}^{2}-\mathrm{a}^{2}\right) \mathrm{m}^{2}-2 \mathrm{x}_{1} \mathrm{y}_{1} \mathrm{~m}+\mathrm{y}_{1}^{2}+\mathrm{b}^{2}=0\). If \(\mathrm{D}<0\), then \(\mathrm{no}\) tangent can be drawn from \(\left(x_{1}, y_{1}\right)\) to the hyperbola.
Note that there are two parallel tangents having the same slope \(\mathrm{m}\).
\(\dfrac{x \sec \theta}{a}-\dfrac{y \tan \theta}{b}=1\)
Note : Point of intersection of the tangents at \(\theta_{1} \) & \(\theta_{2}\) is \(\mathrm{x}=\mathrm{a} \dfrac{\cos \left(\dfrac{\theta_{1}-\theta_{2}}{2}\right)}{\cos \left(\dfrac{\theta_{1}+\theta_{2}}{2}\right)}, \mathrm{y}=\mathrm{b} \tan \left(\dfrac{\theta_{1}+\theta_{2}}{2}\right)\)
8. NORMAL TO THE HYPERBOLA \(\mathbf{\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}\) :
(a) Point form : Equation of the normal to the given hyperbola at the point \(\mathrm{P}\left(\mathrm{x}_{1}, y_{1}\right)\) on it is \(\dfrac{\mathrm{a}^{2} \mathrm{x}}{\mathrm{x}_{1}}+\dfrac{\mathrm{b}^{2} \mathrm{y}}{\mathrm{y}_{1}}=\mathrm{a}^{2}+\mathrm{b}^{2}=\mathrm{a}^{2} \mathrm{e}^{2}\).
(b) Slope form : The equation of normal of slope \(\mathrm{m}\) to the given hyperbola is \(y=m x \mp \dfrac{m\left(a^{2}+b^{2}\right)}{\sqrt{\left(a^{2}-m^{2} b^{2}\right)}}\) foot of normal are \(\left(\pm \dfrac{\mathrm{a}^{2}}{\sqrt{\mathrm{a}^{2}-\mathrm{m}^{2} \mathrm{~b}^{2}}}, \mp \dfrac{\mathrm{mb}^{2}}{\sqrt{\mathrm{a}^{2}-\mathrm{m}^{2} \mathrm{~b}^{2}}}\right)\)
(c) Parametric form :The equation of the normal at the point \(P(a \sec \theta, b \tan \theta)\) to the given hyperbola is \(\dfrac{a x}{\sec \theta}+\dfrac{b y}{\tan \theta}=a^{2}+b^{2}=a^{2} e^{2}\)
9. DIRECTOR CIRCLE :
The locus of the intersection of tangents which are at right angles is known as the Director Circle of the hyperbola. The equation to the director circle is \(: \mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}-\mathrm{b}^{2}\). If \(b^{2}<a^{2}\) this circle is real; if \(b^{2}=a^{2}\) the radius of the circle is zero \(\&\) it reduces to a point circle at the origin. In this case the centre is the only point from which the tangents at right angles can be drawn to the curve.
If \(b^{2}>a^{2}\), the radius of the circle is imaginary, so that there is no such circle & so no tangents at right angle can be drawn to the curve.
10. CHORD OF CONTACT :
If PA and \(\mathrm{PB}\) be the tangents from point \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) to the Hyperbola \(\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\dfrac{y^{2}}{\mathrm{~b}^{2}}=1\), then the equation of the chord of contact \(\mathrm{AB}\) is \(\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}-\dfrac{\mathrm{yy}_{1}}{\mathrm{~b}^{2}}=1\) or \(\mathrm{T}=0 \,\, \mathrm{at}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\)
11. PAIR OR TANGENTS:
where \( S_{1}=\dfrac{x_{1}^{2}}{a^{2}}-\dfrac{y_{1}^{2}}{b^{2}}-1, \quad T=\dfrac{x x_{1}}{a^{2}}-\dfrac{y y_{1}}{b^{2}}-1\)
i.e. \(\left(\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}-1\right)\left(\dfrac{\mathrm{x}_{1}^{2}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y}_{1}^{2}}{\mathrm{~b}^{2}}-1\right)\)\(=\left(\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y} \mathrm{y}_{1}}{\mathrm{~b}^{2}}-1\right)\)
12. EQUATION OF CHORD WITH MID POINT \(\mathbf{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}\) :
whose mid-point be \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) is \(\mathrm{T}=\mathrm{S}_{1}\)
where \(\mathrm{T}=\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}-\dfrac{\mathrm{yy}_{1}}{\mathrm{~b}^{2}}-1, \mathrm{~S}_{1}=\dfrac{\mathrm{x}_{1}^{2}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y}_{1}^{2}}{\mathrm{~b}^{2}}-1\)
i.e. \(\left(\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}-\dfrac{\mathrm{yy}_{1}}{\mathrm{~b}^{2}}-1\right)=\left(\dfrac{\mathrm{x}_{1}^{2}}{\mathrm{a}^{2}}-\dfrac{\mathrm{y}_{1}^{2}}{\mathrm{~b}^{2}}-1\right)\)
13. ASYMPTOTES :
Definition : If the length of the perpendicular let fall from a point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the Hyperbola.
Combined equation of asymptotes of hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) will be \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=0\)
14. RECTANGULAR HYPERBOLA:
Rectangular hyperbola referred to its asymptotes as axis of coordinates.
(a) Equation is \(x y=c^{2}\) with parametric representation \(\mathrm{x}=\mathrm{ct}, \quad \mathrm{y}=\mathrm{c} / \mathrm{t}\) \(\mathrm{t} \, \in \,\mathrm{R}-\{0\}\)
(b) Equation of a chord joining the points \(P\left(t_{1}\right) \) & \(Q\left(t_{2}\right)\) is \(x+t_{1} t_{2} y=c\left(t_{1}+t_{2}\right)\) with slope \(\mathrm{m}=\dfrac{-1}{\mathrm{t}_{1} \mathrm{t}_{2}}\)
(c) Equation of the tangent at \(\mathrm{P}\left(\mathrm{x}_{1}, y_{1}\right)\) is \(\dfrac{\mathrm{x}}{\mathrm{x}_{1}}+\dfrac{\mathrm{y}}{\mathrm{y}_{1}}=2\) & at \(\mathrm{P}(\mathrm{t})\) is \(\dfrac{\mathrm{x}}{\mathrm{t}}+\mathrm{t} y=2 \mathrm{c}\)
(d) Equation of normal is \(y-\dfrac{c}{t}=t^{2}(x-c t)\)
(e) Chord with a given middle point as \((\mathrm{h}, \mathrm{k})\) is \(\mathrm{kx}+\mathrm{hy}=2 \mathrm{hk}\).
15. IMPORTANT HIGHLIGHTS :
(i) The tangent and normal at any point of a hyperbola bisect the angle between the focal radii.
(ii) Reflection property of the hyperbola : An incoming light ray aimed towards one focus is reflected from the outer surface of the hyperbola towards the other focus.
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