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 product of the perpendicular distances from the origin to the pair of straight lines represented by

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

is

\frac{| c |}{\sqrt{(a - b)^{2} + 4 h^{2}}}

.

Accepted Answer

Let, $l_{1} x + m_{1} y + n_{1} = 0$ ...(i) and $l_{2} x + m_{2} y + n_{2} = 0$ ...(ii) be the lines represented by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$

$\Rightarrow a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = (l_{1} x + m_{1} y + n_{1}) (l_{2} x + m_{2} y + n_{2})$

$\Rightarrow l_{1} l_{2} = a, m_{1} m_{2} = b, l_{1} m_{2} + l_{2} m_{1} = 2 h, l_{1} n_{2} + l_{2} n_{1} = 2 g, m_{1} n_{2} + m_{2} n_{1} = 2 f, n_{1} n_{2} = c$

Perpendicular from origin to (i) $= \frac{|n_{1}|}{\sqrt{l_{1}^{2} + m_{1}^{2}}}$

Perpendicular from origin to (ii) $= \frac{|n_{2}|}{\sqrt{l_{2}^{2} + m_{2}^{2}}}$

Product of perpendiculars $= \frac{|n_{1}|}{\sqrt{l_{1}^{2} + m_{1}^{2}}} \cdot \frac{|n_{2}|}{\sqrt{l_{2}^{2} + m_{2}^{2}}}$

$= \frac{|n_{1} n_{2}|}{\sqrt{l_{1}^{2} l_{2}^{2} + m_{1}^{2} m_{2}^{2} + l_{1}^{2} m_{2}^{2} + l_{2}^{2} m_{1}^{2}}}$

$= \frac{|n_{1} n_{2}|}{\sqrt{{(l_{1} l_{2} - m_{1} m_{2})}^{2} + {(l_{1} m_{2} + l_{2} m_{1})}^{2}}}$

$= \frac{| c |}{\sqrt{(a - b)^{2} + (2 h)^{2}}}$

$= \frac{| c |}{\sqrt{(a - b)^{2} + 4 h^{2}}}$

Hence, proved.

Show that the product of the perpendicular distances from the origin to the pair of straight lines represented by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ is $\frac{| c |}{\sqrt{(a - b)^{2} + 4 h^{2}}}$ .

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the product of the perpendicular distances from the origin to the pair of straight lines represented by ax2+2hxy+by2+2gx+2fy+c=0 is |c|(a-b)2+4h2.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the product of the perpendicular distances from the origin to the pair of straight lines represented by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ is $\frac{| c |}{\sqrt{(a - b)^{2} + 4 h^{2}}}$ .