If the pair of lines a x2-2 h x y-b y2-2 g x-2 f y-c-0 intersect on the y axis- thenA- 2 fgh - bg 2- ch 2B- fgh - bg 2C- bg 2- ch 2D- a b c-f g h-a f2-b g2-c h2

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Question

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

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

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$f g h = b g^{2}$

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$b g^{2} = c h^{2}$

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

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Solution

The correct option is A
$2 f g h = b g^{2} + c h^{2}$

$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents straight line (pair)
so $∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow a b c + 2 f g h = a f^{2} + b g^{2} + c h^{2}$
The pair of lines intersect at y axis (x = 0)
So $b y^{2} + 2 f y + c = 0$
$\Rightarrow f^{2} = b c$
So $a b c + 2 f g h = a f^{2} + b g^{2} + c h^{2}$
$\Rightarrow a f^{2} + 2 f g h = a f^{2} + b g^{2} + c h^{2} 2 f g h = b g^{2} + c h^{2}$

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Similar questions

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 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. 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 pair of lines ax2+2hxy+by2+2gx+2fy+c=0 intersect on the y axis, then

The correct option is A 2fgh=bg2+ch2 ax2+2hxy+by2+2gx+2fy+c=0 represents straight line (pair) so ∣∣ ∣∣ahghbfgfc∣∣ ∣∣=0⇒abc+2fgh=af2+bg2+ch2 The pair of lines intersect at y axis (x = 0) So by2+2fy+c=0 ⇒f2=bc So abc+2fgh=af2+bg2+ch2 ⇒af2+2fgh=af2+bg2+ch22fgh=bg2+ch2

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