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

Find the angle between the lines joining the origin to the points of intersection of the curve

x^{2} + 2 x y + y^{2} + 2 x + 2 y - 5 = 0

and the line

3 x - y + 1 = 0

.

Accepted Answer

Equation of the curve is $x^{2} + 2 x y + y^{2} + 2 x + 2 y - 5 = 0$ ...(i)

Equation of $A B$ is $3 x - y + 1 = 0$

$\Rightarrow y - 3 x = 1$ ...(ii)

Let $A, B$ be the points of intersection of the line and the curve.

Homogenizing, (i) with the help of (ii), combined equation of $O A, O B$ is $x^{2} + 2 x y + y^{2} + 2 x \cdot 1 + 2 y \cdot 1 - 5 \cdot 1^{2} = 0$

$\Rightarrow x^{2} + 2 x y + y^{2} + 2 x (y - 3 x) + 2 y (y - 3 x) - 5 (y - 3 x)^{2} = 0$

$\Rightarrow x^{2} + 2 x y + y^{2} + 2 x y - 6 x^{2} + 2 y^{2} - 6 x y - 5 (y^{2} + 9 x^{2} - 6 x y) = 0$

$\Rightarrow - 5 x^{2} - 2 x y + 3 y^{2} - 5 y^{2} - 45 x^{2} + 30 x y = 0$

$\Rightarrow - 50 x^{2} + 28 x y - 2 y^{2} = 0$

$\Rightarrow 25 x^{2} - 14 x y + y^{2} = 0$

Let $θ$ be the angle between $O A$ and $O B$ , then $\cos θ = \frac{| a + b |}{\sqrt{(a - b)^{2} + 4 h^{2}}}$

$= \frac{| 25 + 1 |}{\sqrt{(25 - 1)^{2} + 196}}$

$= \frac{26}{\sqrt{576 + 196}}$

$= \frac{26}{\sqrt{772}}$

$= \frac{26}{2 \sqrt{193}}$

$= \frac{13}{\sqrt{193}}$

$∴ θ = \cos^{- 1} (\frac{13}{\sqrt{193}})$

Find the angle between the lines joining the origin to the points of intersection of the curve $x^{2} + 2 x y + y^{2} + 2 x + 2 y - 5 = 0$ and the line $3 x - y + 1 = 0$ .

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Find the angle between the lines joining the origin to the points of intersection of the curve x2+2xy+y2+2x+2y-5=0 and the line 3x-y+1=0.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Find the angle between the lines joining the origin to the points of intersection of the curve $x^{2} + 2 x y + y^{2} + 2 x + 2 y - 5 = 0$ and the line $3 x - y + 1 = 0$ .