The equation 4x^2 + 12xy + 9y^2 + 2gx + 2fy + | vedclass

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Question

(C) The general equation of second degree $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of parallel lines if $h^2 = ab$ and $af^2 = bg^2$

Vedclass · Accepted Answer

$g = 2, f = 3, c$ is any number

Vedclass · Answer

(C) The general equation of second degree $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of parallel lines if $h^2 = ab$ and $af^2 = bg^2$.Comparing the given equation with the general form,we have $a = 4, h = 6, b = 9$.Check condition $h^2 = ab$: $6^2 = 4 	imes 9 \Rightarrow 36 = 36$,which is satisfied.Check condition $af^2 = bg^2$: $4f^2 = 9g^2$ $\Rightarrow (2f)^2 = (3g)^2$ $\Rightarrow 2f = 3g$ $\Rightarrow f = \frac{3}{2}g$.If we take $g = 2$,then $f = 3$.The condition for the equation to represent a pair of lines is $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Substituting the values: $4 	imes 9 	imes c + 2(3)(2)(6) - 4(3^2) - 9(2^2) - c(6^2) = 0$.$36c + 72 - 36 - 36 - 36c = 0$.$0 = 0$.This identity holds true for any value of $c$. Thus,$g = 2, f = 3$ and $c$ is any number.

The equation $4x^2 + 12xy + 9y^2 + 2gx + 2fy + c = 0$ represents two real parallel straight lines,if

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