Conic Section

Hyperbola

A hyperbola is the locus of a point in a plane such that the difference of its distances from two fixed points is a constant. A conic section whose eccentricity is greater than 1 is a hyperbola.

Standard Equation

Let the two fixed points (called foci) be S(c,0) and S(c,0). The midpoint between the fixed points is the centre of the hyperbola. In this case, the centre is at the origin. Take any point P(x,y) on the hyperbola. Then ifPS>PS,PSPS=ConstantifPS>PS,PSPS=Constant

Standard Equation of a Hyperbola

LetPSPS=2a

By using distance formula, (x+c)2+y2(xc)2+y2=2a (x+c)2+y2=4a2+(xc)2+y2+4a(xc)2+y2 4cx4a2=4a(xc)2+y2 (cxa2)2=a2[(xc)2+y2] a2x2c2x2+a2y2=a4a2c2 x2(a2c2)+a2y2=a2(a2c2) x2a2+y2a2c2=1

Since the difference of two sides of a triangle is less than the third, we have PSPS<SS i.e.2a<2ca<c a2c2is negative.

Let b2=c2a2 then the equation of the hyperbola becomes x2a2y2b2=1

This is the equation of the hyperbola in standard form.

Graph

Graph of Hyperbola

The graph of a hyperbola consists of two branches that go to infinity. The two straight lines y=bax and y=bax are asymptotes to the hyperbola. That is, the distance of a point on the hyperbola from the line tends to zero as the point moves to infinity along the curve.

The vertices of the hyperbola are A(a,0) and A(a,0). The line AA joining the vertices A and A is called the transverse axis and its length is 2a. The line BB perpendicular to the transverse axis such that OB=OB=b is called the conjugate axis and its length is 2b.

The eccentricity e of the hyperbola is defined as e=ca. e=ca=a2+b2a=1+b2a2

Since c>a, e is greater than 1. The coordinates of the foci are (±ae,0). The equation of directrix is x=±ae. Also, the length of the latus rectum is 2b2a.

Conjugate Hyperbola

The hyperbola whose transverse axis is along the y-axis and the conjugate axis along the x-axis is called the conjugate hyperbola. Its equation is y2b2x2a2=1 or,x2a2y2b2=1

Conjugate Hyperbola

Its vertices are A(0,b) and A(0,b). The eccentricity e is given by e=cb=1+a2b2

The coordinates of the foci are (0,±be). The equation of directrix is y=±be. Also, the length of the latus rectum is 2a2b.

Hyperbola whose Asymptotes are the Axes of Coordinates

Hyperbola whose asymptotes are the axes of coordinates

If the foci of a hyperbola be at S(a,a) and S(a,a) and if P(x,y) by any point on the hyperbola, then (x+a)2+(y+a)2(xa)2+(ya)2=2a 4ax+4ay4a2=4a(xa)2+(ya)2 x+ya=(xa)2+(ya)2 (x+ya)2=(xa)2+(ya)2 2xy=a2 or,xy=c2wherec2=a22 xy=c2

The asymptotes of such hyperbola are the axes of coordinates as shown in the above graph.

Equation of the Hyperbola centered at (h,k)

The equation of a hyperbola whose centre is at (h,k) is given by (xh)2a2(yk)2b2=1

Its vertex is (h±a,k) and focus (h±ae,k). The eccentricity e is given by e=1+b2a2

The equation of directrix is x=h±ae and the length of the latus rectum is 2b2a.

Similarly, we can evaluate the hyperbola given by the equation (xh)2a2(yk)2b2=1

Its vertex is (h,k±b) and focus (h,k±be). The eccentricity e is given by e=1+a2b2

The equation of directrix is y=k±be and the length of the latus rectum is 2a2b.


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