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BodhiAI
Nov. 12, 2019

Equation of Angle Bisector of Angle b/w lines Represented by Second Degree Equation & Distance b/w line:

Equation of Angle Bisector(s)

Locus of all such points which are quidistant from two given non-parallel lines is/are the equations of Perpendicular Bisectors between two lines.

Let

and be two equations of non-parallel lines.

Let P(h, k) be any general point lying in the plane containing two lines and such that the distance of this point P from both the lines is equal

i.e.

which are two equations of angle bisectors, one being L and other L' one of them being acute angle bisector and other being an obtuse angle bisector.

Problem Solving Trick(s)

How to Differentiate? Which of the Angle Bisector is Acute angled bisector and which one is obtuse angled bisector?

If

and are any two equations of lines, then the angle between the two lines in parametric from is given by the expression 

Step I Let us first evaluate the value of or the determinant

If 0 i.e. Move to Step II 

else if < 0 i.e.

Step II If , then the equation obtained with positive sign is the bisector of acute angle and the equation obtained with negative sign is the bisector of the obtuse angle.

Also, if , then the equation obtained with POSITIVE sign is the bisector of OBTUSE angle and the equation obtained with NEGATIVE sign will be the bisector of ACUTE angle.

Step III If , then the equation obtained with positive sign is the bisector of acute angle and the equation obtained with negatgive sign is the bisector of the obtuse angle.

Also, if then the equation obtained with POSITIVE sign is the bisector of OBTUSE angle and the equation obtained with NEGATIVE sign will be the bisector of ACUTE angle.

Data Flow Diagram: To Determine The Acute or Obtuse Angle Bisector


Distance Between Parallel Lines Represented by General Equation

Let represents the general equation of pair of lines. Since the lines are parallel to each other 

or

Hence, the general equation of line representing a parallel pair is 

.......(1)

Let the equation of two lines given by above equation be

......(2)

Comparing equations (1) and (2), we have 

pq = c

.......(3)

and pq = c

If and are the equations of parallel lines represented by the two degree general equation, then distance between the parallel lines is given by 

{using (3) & (4)}