If the bisector of the angles between the pairs of lines given by the equation
$a x^{2} + 2 h x y + b y^{2} = 0$ and $a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0$
be coincident, then $λ$ =

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Any real number

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Solution

The correct option is D
Any real number

$Bisectors o f a x^{2} + 2 h x y + b y^{2} = 0 a r e \frac{x^{2} - y^{2}}{a - b} = \frac{x y}{h} . . . . . . (i) and of a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0 i . e ., (a + λ) x^{2} + 2 h x y + (b + λ) y^{2} = 0 a r e \frac{x^{2} - y^{2}}{(a + λ) - (b + λ)} = \frac{x y}{h} \Rightarrow \frac{x^{2} - y^{2}}{a - b} = \frac{x y}{h} . . . . . . . . (i i)$

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If the bisector of the angles between the pairs of lines given by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ and $a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0$ be coincident, then $λ$ =

The lines $a x^{2} + 2 h x y + b y^{2} = 0$ are equally inclined to the lines

$a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0$ for

Q. Prove that angle between one of the lines given by

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

and one of the lines

a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0

is equal to the angle between the other two lines of the system.

Q. If the pair of straight lines

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

is rotated about the origin through

90^{\circ},

then the equations in the new position is

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

represents a pair of straight lines through origin & angle between them is given by

tan θ = \frac{2 \sqrt{h^{2} - a b}}{a + b}

. If the lines are perpendicular then

a + b = 0

and the equation of bisectors is given by

\frac{x^{2} - y^{2}}{a - b} = \frac{x y}{h}

The general equation of second degree given by

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

represent a pair of straight lines if

△ = 0

∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = 0

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

On the basis of above information answer the following question

The Joint equation of the bisectors of the angle between the lines represented by

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

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If the bisector of the angles between the pairs of lines given by the equation ax2+2hxy+by2=0 and ax2+2hxy+by2+λ(x2+y2)=0 be coincident, then λ =

The correct option is D Any real number Bisectors of ax2+2hxy+by2=0 arex2−y2a−b=xyh ......(i)and ofax2+2hxy+by2+λ(x2+y2)=0i.e., (a+λ)x2+2hxy+(b+λ)y2=0 arex2−y2(a+λ)−(b+λ)=xyh⇒ x2−y2a−b=xyh........(ii)

If the bisector of the angles between the pairs of lines given by the equation
$a x^{2} + 2 h x y + b y^{2} = 0$ and $a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0$
be coincident, then $λ$ =