Horizontal Hyperbola
Introduction to Horizontal hyperbola
Introduction to Horizontal hyperbola
A hyperbola is a conic in which eccentricity is greater than unity.Thus a hyperbola is the curve that moves so that the ratio of the distance from a fixed point to its distance from a fixed straight line is greater than 1.The fixed point is called focus,the fixed straight line is called directrix , and the fixed ratio is called eccentricity of the hyperbola.
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NATURE OF THE CURVE
1.The standard form of the hyperbola
s=x2 / a2 - y2 / b2 =1 where a>0 , b>0 and b2 =a2 (e2 -1)
In the equation if we take y=0 then we get x =a
Therefore ,the hyperbola cuts the x-axis at A(a , 0) and A1 =(-a , 0)
In the equation if we put x = 0 then we get y =v(-b2) does not exist in the Cartesian plane.Hence, the curve does not intersect y-axis.
For any value of y , we have two values of x = a / b vy 2+ b2 equal but opposite in sign.
Therefore , the curve is symmetric about the y-axis.
Therefore , the curve consists of two symmetrical branches each extending to infinity in two directions.
'C' is called the center of the hyperbola. It is the point of intersection of the transverse and conjugate axis.It can be shown that 'c' bisects every chord of the hyperbola that passes through itself.
Definition of Rectangular Hyperbola
If in a hyperbola the length of the transverse axis 2a is equal to the length of the conjugate axis 2b the hyperbola is called a RECTANGULAR HYPERBOLA.
Its equation is x2 - y2 =a2 ( a =b )
In this case e2 = a2 +b2 / a2 =2a2 / a2 =2 ( e =v2 )
Therefore, the eccentricity of a rectangular hyperbola is v2.
DEFINITION OF AUXILIARY CIRCLE
The circle described on the transverse axis of a hyperbola as diameter is called the AUXILARY CIRCLE of the hyperbola.
DEFINITION OF CONJUGATE HYPERBOLA
The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis are respectively the conjugate the conjugate and transverse axis of a given hyperbola is called the conjugate hyperbola of the given hyperbola.
The equation of the hyperbola conjugates to
s= x2 / a2 - y2 / b2 =1 is s1 =x2 / a2 - y2 / b2 = -1
For x 2/ a2 - y2 / b2 =1
The transverse axis lies along and its length is 2a.
The conjugate axis lies along y-axis and its length is 2b.
For x2 / a2 -y2 / b2 = -1
The transverse axis lies along y-axis and its length is 2b
The conjugate axis lies along x-axis and its length is 2a.
Therefore , the hyperbola s1=0 is called the conjugate hyperbola of s=0. Also s=0is called the conjugate hyperbola of s1=0.
Thus each is called the conjugate of the other.
Problems on Horizontal Hyperbola
1.Find the equations of the tangents to the hyperbola 3x2 - 4y2 =12 which are parallel and perpendicular to the line y = x-7.
Sol: Equation of the given hyperbola x2 / 4 - y2 / 3 =1 so that a2 =4 , b2 = 3
Equation of the given line is y =x -7 and its slope is 1.
i) Slope of the tangents which are parallel to the given line is m=1
Therefore, equations of tangents are y=mx va2 m2 - b2
=x v4 -3
y =x 1
ii) slope of the tangent which are perpendicular to the given line are
y =( -1)x v4(-1)2 -3
x+y = 1
by: Omkar Nayak
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