Name: Angle Bisectors of Lines
Uploaded: 2019-07-19
Duration: 8 min 21 s
Description: How do you find the equation of an angle bisector of two lines? Well, watch this video to get all the answers related to this! Have fun learning!

Question

Prove that the lines

3 x - 4 y + 5 = 0, x + 2 y = 3, 3 x - 4 y + 15 = 0

and

x + 2 y = 9

form a parallelogram.

Accepted Answer

Given lines $3 x - 4 y + 5 = 0 \to (1), x + 2 y - 3 = 0 \to (2), 3 x - 4 y + 15 = 0 \to (3)$ and $x + 2 y - 9 = 0 \to (4)$ .

We know that, the slope of the general equation of the line $a x + b y + c = 0$ is $- \frac{a}{b}$ . [Since $a x + b y + c = 0 \Rightarrow y = - \frac{a}{b} x - \frac{c}{b}$ , so on comparing with slope intercept form $y = m x + c$ , then we obtain $m = \frac{- a}{b}$ ]

Let $m_{1}$ be the slope of the line $3 x - 4 y + 5 = 0$ .

$∴ m_{1} = \frac{- 3}{- 4} = \frac{3}{4}$

And let $m_{2}$ be the slope of the $x + 2 y = 3$ .

$∴ m_{2} = \frac{- 1}{2} = - \frac{1}{2}$

Also, Let $m_{3}$ be the slope of the line $3 x - 4 y + 15 = 0$ .

$∴ m_{3} = \frac{- 3}{- 4} = \frac{3}{4}$

And let $m_{4}$ be the slope of the line $x + 2 y = 9$ .

$∴ m_{3} = \frac{- 1}{2} = - \frac{1}{2}$

Clearly, the slope of line $3 x - 4 y + 5 = 0$ and slope of the line $3 x - 4 y + 15 = 0$ are equal.

And also, the slope of the line $x + 2 y = 3$ and slope of the line $x + 2 y = 9$ are also equal.

Thus, lines $(1)$ , $(3)$ and $(2)$ , $(4)$ are two pair of parallel lines.

If both pair of opposite sides are parallel then, we can say that it is a parallelogram.
Hence, the given lines form a parallelogram.

Prove that the lines $3 x - 4 y + 5 = 0, x + 2 y = 3, 3 x - 4 y + 15 = 0$ and $x + 2 y = 9$ form a parallelogram.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Prove that the lines 3x-4y+5=0, x+2y=3,3x-4y+15=0 and x+2y=9 form a parallelogram.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Prove that the lines $3 x - 4 y + 5 = 0, x + 2 y = 3, 3 x - 4 y + 15 = 0$ and $x + 2 y = 9$ form a parallelogram.