If an angular bisector of the coordinate axes | vedclass

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(C) The bisectors of the angle between the coordinate axes are $y = x$ and $y = -x$.Since these lines are part of the pair of lines represented by $x^

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(C) The bisectors of the angle between the coordinate axes are $y = x$ and $y = -x$.Since these lines are part of the pair of lines represented by $x^2+2axy+3y^2=0$,they must satisfy the equation.Case $1$: Substitute $y = x$ into the equation:$x^2+2ax(x)+3x^2 = 0$$4x^2+2ax^2 = 0$$x^2(4+2a) = 0$Since this holds for all $x$,$4+2a = 0$,which gives $a = -2$.Case $2$: Substitute $y = -x$ into the equation:$x^2+2ax(-x)+3(-x)^2 = 0$$x^2-2ax^2+3x^2 = 0$$4x^2-2ax^2 = 0$$x^2(4-2a) = 0$Since this holds for all $x$,$4-2a = 0$,which gives $a = 2$.The possible values of $a$ are $-2$ and $2$.Therefore,the sum of all possible values of $a$ is $(-2) + 2 = 0$.

If an angular bisector of the coordinate axes is one of the lines of $x^2+2ax y+3y^2=0$,then the sum of all possible values of $a$ is

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