If the equation
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
represents a pair of straight lines, then
$a b c + 2 f g h - a f^{2} - b g^{2} + c h^{2} = 0$

True

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False

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Solution

The correct option is A
True

$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ is a second degree curve. It can represent a conic or a pair of straight lines. To decide that we define a variable

$△ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$

If $△ = 0$ , it represents a pair of straight lines.

So the given statement is True.

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Similar questions

If the equation
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
represents a pair of straight lines, then
$a b c + 2 f g h - a f^{2} - b g^{2} + c h^{2} = 0$

Q. If

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

represents a parabola, then

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}

is and

If the pair of lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the y axis, then

Q. Assertion :If the equation

6 x^{2} - 42 x y + 60 y^{2} - 11 x + 10 y + λ = 0

represents two straight lines, then

λ = 10.

Reason: The second degree equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

represents a pair of straight lines if

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0

Q. Assertion :The equation

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

represents a hyperbola. Reason: The second degree equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

represents a hyperbola if

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq 0

h^{2} > a b

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If the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of straight lines, then abc + 2fgh − af2 − bg2 + ch2 = 0

If the equation
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
represents a pair of straight lines, then
$a b c + 2 f g h - a f^{2} - b g^{2} + c h^{2} = 0$