If the equation $2 h x y + 2 g x + 2 f y + c = 0$ represents two straight lines, then show that they form a rectangle of area $\frac{| f g |}{h^{2}}$ with the coordinate axes.

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Solution

The given equation $2 h x y + 2 g x + 2 f y + c = 0$
The condition for $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
is $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$\Rightarrow 2 f g h - c h^{2} = 0$
$\Rightarrow 2 f g = c h$
so, $2 h x y + 2 g x + 2 f y + c = 0$
$\Rightarrow 2 h x y + 2 g x + \frac{c h}{x} y + c = 0$
$\Rightarrow 2 x (h y + g) + \frac{c}{g} (h y + g) = 0$
$x = \frac{- c}{2 g}, y = \frac{- g}{h}$
Area of the rectangle
$= | \frac{c}{h} |$
$= | \frac{f g}{h^{2}} |$ units.

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If the equation 2hxy+2gx+2fy+c=0 represents two straight lines, then show that they form a rectangle of area |fg|h2 with the coordinate axes.

The given equation 2hxy+2gx+2fy+c=0The condition for ax2+2hxy+by2+2gx+2fy+c=0is abc+2fgh−af2−bg2−ch2=0⇒2fgh−ch2=0⇒2fg=chso, 2hxy+2gx+2fy+c=0⇒2hxy+2gx+chxy+c=0⇒2x(hy+g)+cg(hy+g)=0x=−c2g,y=−ghArea of the rectangle=|ch|=|fgh2| units.

If the equation $2 h x y + 2 g x + 2 f y + c = 0$ represents two straight lines, then show that they form a rectangle of area $\frac{| f g |}{h^{2}}$ with the coordinate axes.