coordinate Gometry

**rohith14** · September 11th 2009, 06:35 PM

If the eqations of two intersecting straight lines are L1=a1x+b1y+c1=0 &L2=a2x+b2y+c2=0,then eqations of bisectors of angles between L1=0 & L2=0 are
(a1x+b1y+c1)/sqrt(a12 +b12)=+or-(a2x+b2y+c2=0)/sqrt(a22+b22)
(I had a problem of understanding proof of this teorem)

**Grandad** · September 11th 2009, 10:36 PM

Hello rohith14

Originally Posted by rohith14

If the eqations of two intersecting straight lines are L1=a1x+b1y+c1=0 &L2=a2x+b2y+c2=0,then eqations of bisectors of angles between L1=0 & L2=0 are
(a1x+b1y+c1)/sqrt(a12 +b12)=+or-(a2x+b2y+c2=0)/sqrt(a22+b22)
(I had a problem of understanding proof of this teorem)

There are two key facts you need to know in order to understand this theorem:

Any point on the bisector of the angle between two lines is equidistant from the two lines. This can be proved using congruent triangles.

The distance of the point $(x_1, y_1)$ from the line $ax+by+c=0$ is $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$ . The proof of this can be a bit complicated, but there's quite a good one here.

Once you've understood these two facts, the proof of your theorem is very easy. If $(x, y)$ lies on one of the angle bisectors, then its distance from $L_1$ is $\frac{|a_1x+b_1y+c|}{\sqrt{{a_1}^2+{b_1}^2}}$ , and its distance from $L_2$ is $\frac{|a_2x+b_2y+c|}{\sqrt{{a_2}^2+{b_2}^2}}$ . These two distances are equal, and so

$\frac{|a_1x+b_1y+c|}{\sqrt{{a_1}^2+{b_1}^2}}=\frac {|a_2x+b_2y+c|}{\sqrt{{a_2}^2+{b_2}^2}}$

$\Rightarrow\frac{a_1x+b_1y+c}{\sqrt{{a_1}^2+{b_1}^ 2}}=\pm\frac{a_2x+b_2y+c}{\sqrt{{a_2}^2+{b_2}^2}}$

Grandad

**rohith14** · September 14th 2009, 03:22 AM

okay.I am clear now. But i can't understand how can we make answer by solving ax1+by1+c1/........?

**Grandad** · September 14th 2009, 06:22 AM

Hello rohith14

Originally Posted by rohith14

okay.I am clear now. But i can't understand how can we make answer by solving ax1+by1+c1/........?

I think you aren't understanding the basic principle of how we find the equation of a particular locus - the path of a point that moves according to some rule (or rules). This understanding is fundamental to the whole of Cartesian geometry, so read what I've written here very carefully.

The equation of a locus is the relationship between two variables, $x$ and $y$ , which represent the coordinates of a point in the Cartesian plane. Any point $(x, y)$ whose coordinates satisfy that equation lie on the locus; those points whose coordinates do not satisfy the equation do not lie on the locus.

So to find the equation of a particular locus, then, we consider a general point $(x,y)$ , and write down (if we can!) an equation that connects $x$ and $y$ whenever the point $(x,y)$ satisfies the condition(s) that define the locus. Once we've done that, we have found the equation of the locus. It's as simple as that!

In the example of the angle bisectors of the lines $L_1$ and $L_2$ , the defining property of the locus that I have used is this:

It is the path of a point that moves so that its distances from the two fixed lines $L_1$ and $L_2$ are equal.

So we simply write down these distances in terms of $x$ and $y$ , and equate them. And that's it! That is the equation of the locus: the equation of the angle bisectors.

I hope that helps to clarify what is essentially the very essence of coordinate geometry.

Grandad

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