Angle bisector of paired lines

Jul 2009
338
14
Singapore
By considering the set of points P(x, y) that are equidistant from the two lines represented by the equation \(\displaystyle ax^2+2hxy+by^2=0\) show that the equation of the pair of angle bisectors of these lines is
\(\displaystyle hx^2+(b-a)xy-hy^2=0\)

I know that the pair of lines that are bisectors are perpendicular to each other and all the lines intersect at the origin. I don't know what else, or how to begin.
Thanks!
 
Jun 2009
806
275
The pair of line can be written as

\(\displaystyle y - m_1x = 0\) and \(\displaystyle y - m_2x = 0\)

where \(\displaystyle m_1 + m_2 = -\frac{2h}{b}\) and \(\displaystyle m_1m_2 = \frac{a}{b}\)

Equations of angle bisectors are given by

\(\displaystyle \frac{y - m_1x}{\sqrt{1+m_1^2}} - \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0\)

\(\displaystyle \frac{y - m_1x}{\sqrt{1+m_1^2}} + \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0\)

The combined equation of the angle bisectors is given by

\(\displaystyle \frac{(y - m_1x)^2}{1+m_1^2} - \frac{(y - m_2x)^2}{1+m_2^2}= 0\)

Simplify and substitute the values of m1+m2 and m1m2.
 
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Jul 2009
338
14
Singapore
umm how did you get the two first expressions? i don't understand
 
Jun 2009
806
275
umm how did you get the two first expressions? i don't understand
Equation of a straight line passing through the origin is given by

y = mx, or y - mx = 0. If m1 and m2 are the slopes of a pair of straight lines, the combined equation is given by

\(\displaystyle (y - m_1x)(y-m_2x) = 0\)

\(\displaystyle y^2 - (m_1+m_2)x + m_1m_2x^2 = 0\)

Compare this equation with equation given in the problem. You can see that

\(\displaystyle m_1 + m_2 = -\frac{2h}{b}\)

and \(\displaystyle m_1m_2 = \frac{a}{b}\)
 
Jul 2009
338
14
Singapore
oops, sorry i meant these:
\(\displaystyle \frac{y - m_1x}{\sqrt{1+m_1^2}} - \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0\)
\(\displaystyle \frac{y - m_1x}{\sqrt{1+m_1^2}} + \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0\)
thanks!
 
Jun 2009
806
275
oops, sorry i meant these:

thanks!
The angle bisector is the locus of points which are equidistant from the two straight lines.

The two equations are two angle bisectors obtained by finding the perpendicular distance from a point (x, y)
 
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