If ax^2-xy-3y^2-5x+20y+c=0 represents a pair | vedclass

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(B) The equation $ax^2-xy-3y^2-5x+20y+c=0$ represents a pair of lines passing through $(2,3)$.Substituting $(2,3)$ into the equation:$a(2)^2 - (2)(3)

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$27$

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(B) The equation $ax^2-xy-3y^2-5x+20y+c=0$ represents a pair of lines passing through $(2,3)$.Substituting $(2,3)$ into the equation:$a(2)^2 - (2)(3) - 3(3)^2 - 5(2) + 20(3) + c = 0$$4a - 6 - 27 - 10 + 60 + c = 0$$4a + c = -17$ ...$(1)$For the equation to represent a pair of lines,the determinant of the matrix of the quadratic form must be zero:$\begin{vmatrix} a & -1/2 & -5/2 \ -1/2 & -3 & 10 \ -5/2 & 10 & c \end{vmatrix} = 0$$a(-3c - 100) + 1/2(-c/2 + 25) - 5/2(-5 - 15/2) = 0$$-3ac - 100a - c/4 + 25/2 + 25/2 + 75/4 = 0$$3ac + 100a + c/4 = 175/4$Substituting $c = -17 - 4a$ from $(1)$:$12a(-17 - 4a) + 400a + (-17 - 4a) = 175$$-204a - 48a^2 + 400a - 17 - 4a = 175$$-48a^2 + 192a - 192 = 0$$a^2 - 4a + 4 = 0$ $\Rightarrow (a-2)^2 = 0$ $\Rightarrow a = 2$Substituting $a=2$ into $(1)$: $c = -17 - 4(2) = -25$Therefore,$a - c = 2 - (-25) = 27$.

If $ax^2-xy-3y^2-5x+20y+c=0$ represents a pair of lines passing through the point $(2,3)$,then $a-c=$

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