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

Through the point

(3, 4)

are drawn two straight lines each inclined at

45 °

to the straight line

x - y = 2

. Find their equations and find also the area included by the three lines.

Accepted Answer

Let $C$ be the given point and let the given straight line be $P A B$ , making an angle $α$ with the axis of $x$ such that

$m = \tan α$ and $c = (3, 4)$

Equation of line is given $x - y = 2 . . . (i)$

$\Rightarrow y = x - 2$

So, $m = 1$

i.e., $\tan α = 1$

$\Rightarrow α = 45 °$

In $∆ P A Q$ [from the figure]

$∠ A P Q = 45 °$ and $∠ P A Q = 45 °$

So, $∠ A Q P = 90 °$ which is slope of line $C Q$

$∴$ Equation of required line $C Q$ passes through $C (3, 4)$ and makes $90 °$ with positive direction of $x$ -axis is

$y - y_{1} = m (x - x_{1})$

$\Rightarrow y - 4 = \frac{1}{0} (x - 3)$

$\Rightarrow x = 3 . . . (ii)$

And, also from figure, if

$∠ C A B = ∠ C B A = 45 °$

Then line $C B$ will be perpendicular to line $C A Q$

i.e., slope of $C B = 0$

Hence, equation of line $C B$

$(y - 4) = 0 (x - 3)$

$\Rightarrow y - 4 = 0$

$\Rightarrow y = 4 . . . (iii)$

Point of intersection of $(ii) and (iii)$ is $A (3, 4)$

Point of intersection of $(i) and (ii)$ is $B (3, 1)$

Point of intersection of $(i) and (iii)$ is $C (6, 4)$

Area of triangle $A B C = Absolute value of \frac{1}{2} |\begin{matrix} 3 - 3 & 3 - 6 \\ 4 - 1 & 4 - 4 \end{matrix}| = \frac{9}{2} sq . units$

Through the point $(3, 4)$ are drawn two straight lines each inclined at $45 °$ to the straight line $x - y = 2$ . Find their equations and find also the area included by the three lines.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Through the point (3,4) are drawn two straight lines each inclined at 45° to the straight line x-y=2. Find their equations and find also the area included by the three lines.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Through the point $(3, 4)$ are drawn two straight lines each inclined at $45 °$ to the straight line $x - y = 2$ . Find their equations and find also the area included by the three lines.