Two of the lines represented by the equation | vedclass

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Question

(A) Let the equation be $a y^4+b x y^3+c x^2 y^2+d x^3 y+e x^4 = 0$. Assume the equation can be factored as $(a x^2+p x y-a y^2)(x^2+q x y+y^2) = 0$.

Vedclass · Accepted Answer

$(b+d)(a d+b e)+(e-a)^2(a+c+e)=0$

Vedclass · Answer

(A) Let the equation be $a y^4+b x y^3+c x^2 y^2+d x^3 y+e x^4 = 0$. Assume the equation can be factored as $(a x^2+p x y-a y^2)(x^2+q x y+y^2) = 0$. Comparing the coefficients of similar terms: $b = aq - p$,$c = -pq$,$d = aq + p$,and $e = -a$. Then,$b + d = 2aq$ and $e - a = -2a$. Also,$ad + be = 2ap$ and $a + c + e = -pq$. Now,consider the expression $(b+d)(ad+be) = (2aq)(2ap) = 4a^2pq$. Also,$-(e-a)^2(a+c+e) = -(-2a)^2(-pq) = 4a^2pq$. Therefore,$(b+d)(ad+be) = -(e-a)^2(a+c+e)$. This simplifies to $(b+d)(ad+be) + (e-a)^2(a+c+e) = 0$.

Two of the lines represented by the equation $a y^4+b x y^3+c x^2 y^2+d x^3 y+e x^4=0$ will be perpendicular,then

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