Name: Finding the Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 9 min 3 s
Description: In this video, learn the angle between pair of lines and the nature of lines along with examples on them, which help you solve the questions in exams. Peruse and learn.

Question

Show that the lines represented by

{(l x + m y)}^{2} - 3 {(m x - l y)}^{2} = 0

and

l x + m y + n = 0

form an equilateral triangle with area

\frac{n^{2}}{\sqrt{3} (t^{2} + m^{2})}

.

Accepted Answer

Equation of the pair of lines is $(l x + m y)^{2} - 3 (m x - l y)^{2} = 0$ ...(i)

$\Rightarrow l^{2} x^{2} + m^{2} y^{2} + 2 l m x y - 3 m^{2} x^{2} - 3 l^{2} y^{2} + 6 l m x y = 0$

$\Rightarrow (l^{2} - 3 m^{2}) x^{2} + 8 l m x y + (m^{2} - 3 l^{2}) y^{2} = 0$

The point of intersection of above lines is $O (0, 0)$ .

Let angle between pair of lines $a x^{2} + 2 h x y + b y^{2} = 0$ is given by $θ$ , then $\cos θ = \frac{| a + b |}{\sqrt{(a - b)^{2} + 4 h^{2}}}$ .

Let $θ$ be the angle between the lines, then $\cos θ$ $= \frac{|l^{2} - 3 m^{2} + m^{2} - 3 l^{2}|}{\sqrt{{(l^{2} - 3 m^{2} - m^{2} + 3 l^{2})}^{2} + 64 l^{2} m^{2}}}$

$= \frac{2 |l^{2} + m^{2}|}{4 \sqrt{{(l^{2} - m^{2})}^{2} + 4 l^{2} m^{2}}}$

$= \frac{2 |l^{2} + m^{2}|}{4 (l^{2} + m^{2})}$

$\Rightarrow \cos θ = \frac{1}{2}$

$\Rightarrow θ = 60 °$

Given, $(l x + m y)^{2} - 3 (m x - l y)^{2} = 0$

$\Rightarrow (l x + m y)^{2} - (\sqrt{3} (m x - l y))^{2} = 0$

$\Rightarrow ((l x + m y) - \sqrt{3} (m x - l y)) ((l x + m y) + \sqrt{3} (m x - l y)) = 0$

$\Rightarrow (l x + m y) - \sqrt{3} (m x - l y) = 0$ and $(l x + m y) + \sqrt{3} (m x - l y) = 0$

$\Rightarrow (l - m \sqrt{3}) x + (m + \sqrt{3} l) y = 0$ ...(ii) and $(l + m \sqrt{3}) x + (m - \sqrt{3} l) y = 0$ ...(iii)

Equation of given line is $l x + m y + n = 0$ ...(iv)

Let the angle between (ii) and (iv) be $α$ , then $\cos α = \frac{l (l + \sqrt{3} m) + m (m - \sqrt{3} l)}{\sqrt{(l + \sqrt{3} m)^{2} + (m - \sqrt{3} l)^{2}} \sqrt{l^{2} + m^{2}}}$

$\Rightarrow \cos α = \frac{l^{2} + m^{2}}{\sqrt{4 (l^{2} + m^{2})} \sqrt{l^{2} + m^{2}}}$

$\Rightarrow \cos α = \frac{1}{2}$

$\Rightarrow α = 60 °$

Now angle between (iii) and (iv) $= 180 ° - (60 + 60)$

$= 60 °$

Therefore, the angles of the triangle are $60 °, 60 °, 60 °$

Hence, the triangle is an equilateral triangle

Let, $p =$ Length of the perpendicular from $p$ to line $l x + m y + n = 0 = \frac{| n |}{\sqrt{l^{2} + m^{2}}}$

$∴$ Area of $Δ O A B = \frac{p^{2}}{\sqrt{3}}$

$= \frac{n^{2}}{\sqrt{3} (l^{2} + m^{2})}$

Show that the lines represented by ${(l x + m y)}^{2} - 3 {(m x - l y)}^{2} = 0$ and $l x + m y + n = 0$ form an equilateral triangle with area $\frac{n^{2}}{\sqrt{3} (t^{2} + m^{2})}$ .

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the lines represented by lx+my2-3mx-ly2=0 and lx+my+n=0 form an equilateral triangle with area n23t2+m2.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the lines represented by ${(l x + m y)}^{2} - 3 {(m x - l y)}^{2} = 0$ and $l x + m y + n = 0$ form an equilateral triangle with area $\frac{n^{2}}{\sqrt{3} (t^{2} + m^{2})}$ .