Question

Find the separate equations of the lines represented by the equation

12x^2+xy-6y^2+25x-11y+7=0

`12 {x}^(2) + xy - 6 {y}^(2) + 25x - 11 + 7 = 0`
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Answer 1

Hello,

Explanation:

Let's find the center of the conic.

`f(x,y)=12x^2+xy-6y^2+25x-11y+7=0\\\\\left\{\begin{array}{c}\frac{\partial{f}}{\partial{x}} =24x+y+25=0\\\\\frac{\partial{f}}{\partial{y}} =x-12y-11=0\\\end {array} \right.\\\\\\\left\{\begin{array}{c}x=-1\\y=-1\\\end {array} \right.\\\\\\center(-1,-1)\\`

Let find 2 other points of the conic:

if x=2 then y=-5 : A(2,-5)

if x=-3 then y=-4: B(-3,4)

`f(x,y)=(4x+3y+7)(3x-2y+1)=0\\`