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Standard XII

Mathematics

Perpendicular Distance of a Point from a Line

If θ is the...

Question

If $θ$ is the measure of the acute angle between the lines represented by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ , then prove that $tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣$ where $a + b \neq 0$ and $b \neq 0$ .
Find the condition for coincident lines

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Solution

If angle between both the line is $0^{o}$ , then both the lines will be coincident
$\Rightarrow tan θ = 0$

$\Rightarrow \frac{2 \sqrt{h^{2} - a b}}{a + b} = 0 \Rightarrow h^{2} = a b$

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Similar questions

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

Q. Assertion :If the angle between the two lines represented by

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

{tan}^{- 1} (m)

, then value of

m = \frac{1}{5}

. Reason: The angle

θ

between the two lines

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

is calculated by

{tan}^{- 1} \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

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 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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If θ is the measure of the acute angle between the lines represented by the equation ax2+2hxy+by2=0, then prove that tanθ=∣∣∣2√h2−aba+b∣∣∣ where a+b≠0 and b≠0.Find the condition for coincident lines

If angle between both the line is 0o, then both the lines will be coincident ⇒tanθ=0⇒2√h2−aba+b=0⇒h2=ab

If $θ$ is the measure of the acute angle between the lines represented by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ , then prove that $tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣$ where $a + b \neq 0$ and $b \neq 0$ .
Find the condition for coincident lines

If angle between both the line is $0^{o}$ , then both the lines will be coincident
$\Rightarrow tan θ = 0$

$\Rightarrow \frac{2 \sqrt{h^{2} - a b}}{a + b} = 0 \Rightarrow h^{2} = a b$