Name: Condition for Concurrency of Three Lines
Uploaded: 2019-07-19
Duration: 10 min 26 s
Description: Well, this video beautifully explains the conditions for concurrency of three lines with some interesting examples. Come, let's learn and watch the video...!!

Question

Starting at the origin a beam of light hits a mirror (in the form of a line) at the point

A (4, 8)

and reflected line passes through the point

B (8, 12) .

Compute the slope of the mirror.

Accepted Answer

Let the slope of incident line $O A$ be $m_{1}$ and slope of reflected line $B (8, 12) .$ be $m_{2}$ , where $O (0, 0), A (4, 8) & B (8, 12)$ are the given points.

Slope of line $O A = m_{1} = \frac{8 - 0}{4 - 0} = 2$

and slope of line $A B = m_{2} = \frac{12 - 8}{8 - 4} = 1$

Since we know that incident angle and reflected angle are equal.

Let slope of mirror be $m$ ,

$\Rightarrow$ Angle between line $O A$ and mirror $=$ Angle between line $A B$ and mirror

$\Rightarrow \tan^{- 1} |\frac{m_{1} - m}{1 + m m_{1}}| = \tan^{- 1} |\frac{m - m_{2}}{1 + m m_{2}}|$

$\Rightarrow \frac{2 - m}{1 + 2 m} = \pm \frac{m - 1}{1 + m}$

If $\frac{2 - m}{1 + 2 m} = - \frac{m - 1}{1 + m}$

$\Rightarrow (2 - m) (1 + m) = - (m - 1) (1 + 2 m)$

$\Rightarrow - m^{2} + m + 2 = - (2 m^{2} - m - 1)$

$\Rightarrow m^{2} = - 1$ , which is not possible.

If $\frac{2 - m}{1 + 2 m} = \frac{m - 1}{1 + m}$

$\Rightarrow (2 - m) (m + 1) = (m - 1) (1 + 2 m)$

$\Rightarrow - m^{2} + m + 2 = 2 m^{2} - m - 1$

$\Rightarrow 3 m^{2} - 2 m - 3 = 0$

$m = \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 \times 3 \times (- 3)}}{2 \times 3}$

$\Rightarrow m = \frac{2 \pm \sqrt{4 + 36}}{6} = \frac{2 \pm \sqrt{40}}{6}$

$\Rightarrow m = \frac{1 \pm \sqrt{10}}{3}$

Since $m = \frac{1 - \sqrt{10}}{3}$ (negative) is not admissible.

So the slope of mirror $= m = \frac{1 + \sqrt{10}}{3}$

Starting at the origin a beam of light hits a mirror (in the form of a line) at the point $A (4, 8)$ and reflected line passes through the point $B (8, 12) .$ Compute the slope of the mirror.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Starting at the origin a beam of light hits a mirror (in the form of a line) at the point A(4, 8) and reflected line passes through the point B(8,12). Compute the slope of the mirror.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Starting at the origin a beam of light hits a mirror (in the form of a line) at the point $A (4, 8)$ and reflected line passes through the point $B (8, 12) .$ Compute the slope of the mirror.