Question; Find the two straight lines represented by the equations:

a) 2x² +xy -6y² -16x +24y = 0
Solution:
Given,

Single equation of line is:
$2x² +xy -6y² -16x +24y = 0$
$or, 2x² + (4-3)xy -6y² -8(2x -3y) = 0$
$or, 2x² +4xy -3xy -6y² - 8(2x -3y) = 0$
$or, 2x(x +2y) -3y(x +2y) -8(2x -3y) = 0$
$or, (2x -3y)(x +2y) - 8(2x -3y) = 0$
$or, (2x -3y)(x +2y -8) = 0$

So, equation of line 1 is $2x -3y = 0$
And, equation of line 2 is $x +2y -8 = 0$

Therefore, the required separate equations of lines are (2x -3y = 0) and (x +2y -8 = 0).


c) 6x² -xy -12y² -8x +12y = 0
Solution:
Given,

Single equation of line is:
$6x² - xy -12y² -8x +12y = 0$
$or, 6x² - (9-8)xy -12y² -8x +12y = 0$
$or, 6x² -9xy +8xy -12y² -8x +12y = 0$
$or, 3x(2x -3y) + 4y(2x -3y) -4(2x -3y) = 0$
$or, (3x +4y)(2x -3y) - 4(2x -3y) = 0$
$or, (2x -3y)(3x +4y -4) = 0$

So, equation of line 1 is $2x -3y = 0$
And, equation of line 2 is $3x +4y -4 = 0$

Therefore, the required separate equations of lines are (2x -3y = 0) and (3x +4y -4 = 0).

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