Coordinate Geometry - Equation of Lines with respect to other lines

We have already seen the different ways in which we can write the equation of a single line. In this article, we will learn how to write equations of multiple lines.

Equation of a line with respect to another line

Equations of a straight line that passes through a point A ( $x_{1}$ , $y_{1}$ ) and makes an angle of θ with the line y = mx + c are: Coordinate Geometry

$y - y_{1} = \frac{m \pm t a n θ}{1 \mp m t a n θ} (x - x_{1})$

Equation of a line with respect to two other lines

Equation of a line pssing through intersection point

A line that passes through the point of intersection of the lines $a_{1} x + b_{1} y + c_{1}$ = 0 and $a_{2} x + b_{2} y + c_{2}$ = 0, can be represented by the following equation:

$(a_{1} x + b_{1} y + c_{1}) + λ (a_{2} x + b_{2} y + c_{2}$ ) = 0, where λ is a constant.

Point of intersection of two lines

If we have two lines $a_{1} x + b_{1} y + c_{1}$ = 0 and $a_{2} x + b_{2} y + c_{2}$ = 0, then:

$\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

Thus, point of intersection of these two lines = $\frac{b_{1} c_{2} - b_{2} c_{1}}{a_{1} b_{2} - a_{2} b_{1}}, \frac{c_{1} a_{2} - c_{2} a_{1}}{a_{1} b_{2} - a_{2} b_{1}}$
Where $a_{1} b_{2} - a_{2} b_{1}$ ≠ 0

Equations of Angle Bisectors

The equations of the angle bisectors of two line $a_{1} x + b_{1} y + c_{1}$ = 0 and $a_{2} x + b_{2} y + c_{2}$ = 0, can be represented as follows: Coordinate Geometry

$\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \pm \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

How to determine which bisector equation is of Acute and Obtuse angle?

Now, let us see how to find:

whether the origin lies in the acute angle or obtuse angle between the lines.
which bisector equation corresponds to acute angle bisector and obtuse angle bisector.

Step I: Ensure that both $c_{1}$ and $c_{2}$ are positive. If one of them, or both are negative, then make them positive by multiplying both the sides of the concerned equation by -1.

Step II: Find out the sign of $a_{1} a_{2} + b_{1} b_{2}$ .

If $a_{1} a_{2} + b_{1} b_{2}$ > 0, then:

the origin lies in the obtuse angle, and
the “+” symbol gives the bisector of the obtuse angle. That is, Equation of the bisector of the obtuse angle will be: $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

If $a_{1} a_{2} + b_{1} b_{2}$ < 0, then:

the origin lies in the acute angle, and
the “+” symbol gives the bisector of the acute angle. That is, Equation of the bisector of the acute angle will be: $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

Angle between two lines

If θ is the angle between two lines y = $m_{1} x + c_{1}$ and y = $m_{2} x + c_{2}$ , then:
tan θ = $| \frac{m_{2} - m_{1}}{1 + m_{2} m_{1}} |$ or $| \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$

If θ is the angle between two lines $a_{1} x + b_{1} y + c_{1}$ = 0 and $a_{2} x + b_{2} y + c_{2}$ = 0, then:
tan θ = $\frac{a_{2} b_{1} - a_{1} b_{2}}{a_{1} a_{2} + b_{1} b_{2}}$

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Last updated on Sep 27, 2021