The angle between the lines represented by the equation ax 2-2 hxy - by 2-0 is given byA- tan-2h2-a b-a-bB- tan-2 -h2-a b-a-bC- tan-2h2-a b-a-bD- tan-2 -h2-a b-a-b

The correct option is B It is a formula. nbsp; ...

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Standard XII

Mathematics

Direction Cosines

The angle bet...

Question

The angle between the lines represented by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ is given by

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Solution

The correct option is B

It is a formula.

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Tangent of acute angle between pair of straight lines $a x^{2} + 2 h x y + b y^{2}$ is given by $t a n θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a - b} ∣ ∣ ∣$

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

Q. Statement A: The point of intersection of the lines represented by

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

(\frac{b g - h f}{h^{2} - a b}, \frac{a f - g h}{h^{2} - a b})

Statement B: The point of intersection of

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

(\sqrt{(\frac{f^{2} - b c}{h^{2} - a b})}, \sqrt{(\frac{g^{2} - a c}{h^{2} - a b})})

Q. If

θ

is the acute angle between the lines represented by equation

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

, then prove that

tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣, a + b \neq 0

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

If the angle between the lines represented by

6 x^{2} + 5 x y - 4 y^{2} + 7 x + 13 y - 3 = 0

{tan}^{- 1} (m)

a^{2} + b^{2} - a b - a - b + 1 \leq 0

, then

5 a + 6 b

is equal to

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The angle between the lines represented by the equation ax2+2hxy+by2=0 is given by

The correct option is B It is a formula.

The angle between the lines represented by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ is given by

The correct option is B

It is a formula.