Question:

Equation of the bisector of the acute angle between lines $3x + 4y + 5 = 0$ and $12x -5y - 7 = 0$ is

Updated On: Jun 18, 2022
  • $21x + 77y + 100 = 0$
  • $99x - 27y+ 30= 0$
  • $99x + 27y+ 30= 0$
  • $21x - 77y + 100 = 0$
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The Correct Option is C

Solution and Explanation

Given equations are
$3 x+4 y+5=0$ and $12 x-5 y-7=0$
$\therefore a_{1} a_{2}+b_{1} b_{2}=3 \times 12+4 \times(-5)$
$=16>0$
$\therefore$ For acute angle bisector
$\frac{ a _{1} x + b _{1} y + c _{1}}{\sqrt{ a _{1}^{2}+ b _{1}^{2}}}=-\frac{\left( a _{2} x + b _{2} y + c _{2}\right)}{\sqrt{ a _{2}^{2}+ b _{2}^{2}}} $
$\therefore \frac{3 x +4 y +5}{\sqrt{9+16}}=-\frac{(12 x -5 y -7)}{\sqrt{12^{2}+(-5)^{2}}} $
$\Rightarrow \frac{3 x +4 y +5}{5}=-\frac{(12 x -5 y -7)}{13}$
$\Rightarrow 39 x +52 y +65=-60 x +25 y +35 $
$\Rightarrow 99 x +27 y +30=0$
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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c