Question

The equation of the pair of bisectors of the angles between two straight lines is, 12x^2 - 7xy - 12y^2 = 0. if the equation of one line is 2y - x = 0 then the equation of other line is?

Answer 1

Let the pair of intersecting lines be represented by: ax² +2hxy + by² = 0
Then the pair of bisectors of the angles between them is :
           h (x² - y²) = (a - b) x y

we are given: pair of bisectors:  12 (x² - y²) = 7 xy
        so  h = 12   and   (a-b) = 7

So pair of intersecting lines:  (7+b) x² + 24 xy + b y² = 0
one of the lines is   2 y - x = 0
so          (2 y - x) * [ b/2 y - (7+b) x ] = (7+b) x² + 24 xy + b y²
       
equating coefficients:  - b/2 - 2(7+b) = 24
                                =>    b = -76/5
                               so  a = b+7= -41/5
so the other line : b/2 y - (7+b) x = 0
                             -38/5  y  +41/5 x = 0
                                  38 y - 41 x = 0

    pair of intersecting lines :  41 x² - 120 xy + 76 y² = 0

Answer 2

Answer:

Step-by-step explanation:

Let the pair of intersecting lines be represented by: ax² +2hxy + by² = 0

Then the pair of bisectors of the angles between them is :

          h (x² - y²) = (a - b) x y

we are given: pair of bisectors:  12 (x² - y²) = 7 xy

       so  h = 12   and   (a-b) = 7

So pair of intersecting lines:  (7+b) x² + 24 xy + b y² = 0

one of the lines is   2 y - x = 0

so          (2 y - x) * [ b/2 y - (7+b) x ] = (7+b) x² + 24 xy + b y²

       

equating coefficients:  - b/2 - 2(7+b) = 24

                               =>    b = -76/5

                              so  a = b+7= -41/5

so the other line : b/2 y - (7+b) x = 0

                            -38/5  y  +41/5 x = 0

                                 38 y - 41 x = 0

  pair of intersecting lines :  41 x² - 120 xy + 76 y² = 0