Equation of a hyperbola given its asymptotes Find the equation of the hyperbola whose asymptotes are 3x−4y+7 and 4x+3y+1=0 and which pass through the origin. The equation of the hyperbola is obtained in my reference as (3x−4y+7)(4x+3y+1)=K=7

Shyla Odom

Shyla Odom

Open question

2022-08-25

Equation of a hyperbola given its asymptotes
Find the equation of the hyperbola whose asymptotes are 3 x 4 y + 7 and 4 x + 3 y + 1 = 0 and which pass through the origin.
The equation of the hyperbola is obtained in my reference as
( 3 x 4 y + 7 ) ( 4 x + 3 y + 1 ) = K = 7
So it make use of the statement, the equation of the hyperbola = equation of pair of asymptotes + constant
I understand that the pair of straight lines is the limiting case of hyperbola.
Why does the equation to the hyperbola differ from the equation of pair of asymptotes only by a constant ?

Answer & Explanation

Dabbaghnn

Dabbaghnn

Beginner2022-08-26Added 6 answers

Consider the equation of a hyperbola
(1) ( x x 0 ) 2 a 2 ( y y 0 ) 2 b 2 = 1
Which has its asymptotes
( y y 0 ) = ± b a ( x x 0 )
Upon multioplication of the equations of the two asymptotes we get
( y y 0 ) 2 = b 2 a 2 ( x x 0 ) 2
or
(2) ( x x 0 ) 2 a 2 ( y y 0 ) 2 b 2 = 0
As you see the difference of (1) and (2) is a constant.

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