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

Let

A B C

be a triangle such that the coordinates of the vertex

A

are

(- 3, 1)

. Equation of the median through

B

is

2 x + y - 3 = 0

and equation of the angular bisector of

C

is

7 x - 4 y - 1 = 0

. Find the slope of line

B C .

Accepted Answer

Given equation of angular bisector of point $C$ be $7 x - 4 y - 1 = 0$ . So, let $C D$ be angular bisector of point $C$ .

Consider the coordinates of $C$ as $(h, k)$ , and since the point $C$ lies on the angular bisector of $C$ , therefore the equation becomes,

$7 h - 4 k = 1 \dots (1)$

Since the median through point $B$ is $B E$ which is $2 x + y - 3 = 0$

The midpoint of $A C$ will be $E (\frac{h - 3}{2}, \frac{k + 1}{2})$

Hence, $2 (\frac{h - 3}{2}) + \frac{k + 1}{2} - 3 = 0$

$\Rightarrow 2 h + k = 11 \dots (2)$

From equation $(1) & (2)$ ,

$h = 3, k = 5$

So, Slope of line $A C$ is $= \frac{5 - 1}{3 - (- 3)} = \frac{2}{3}$

The slope of bisector $C D$ $(7 x - 4 y - 1 = 0)$ $= \frac{7}{4}$

So, from diagram we can see that the angle between $A C$ and the bisector is equal to angle between $B C$ and the bisector.

Let the angle be $θ$ and slope of $B C$ be $m$ .

$\Rightarrow \tan θ = |\frac{\frac{2}{3} - \frac{7}{4}}{1 + (\frac{2}{3}) (\frac{7}{4})}| = |\frac{m - \frac{7}{4}}{1 + \frac{7}{4} m}|$

$\Rightarrow |\frac{4 m - 7}{4 + 7 m}| = \frac{1}{2}$ $\Rightarrow \frac{4 m - 7}{4 + 7 m} = \pm \frac{1}{2}$

If $\frac{4 m - 7}{4 + 7 m} = \frac{1}{2}$ , then $m = 18$

If $\frac{4 m - 7}{4 + 7 m} = - \frac{1}{2}$ , then $m = \frac{2}{3}$

If $m = \frac{2}{3}$ , then slope of $B C & A C$ will be equal and the points $A, B, C$ will be collinear and it will form no triangle.

Therefore, slope of $B C$ is $18$ .

Let $A B C$ be a triangle such that the coordinates of the vertex $A$ are $(- 3, 1)$ . Equation of the median through $B$ is $2 x + y - 3 = 0$ and equation of the angular bisector of $C$ is $7 x - 4 y - 1 = 0$ . Find the slope of line $B C .$

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Let ABC be a triangle such that the coordinates of the vertex A are (-3,1) . Equation of the median through B is 2 x+y-3=0 and equation of the angular bisector of C is 7x-4y-1=0. Find the slope of line BC.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

Let $A B C$ be a triangle such that the coordinates of the vertex $A$ are $(- 3, 1)$ . Equation of the median through $B$ is $2 x + y - 3 = 0$ and equation of the angular bisector of $C$ is $7 x - 4 y - 1 = 0$ . Find the slope of line $B C .$