# Thread: Product & Sum of Roots

1. ## Product & Sum of Roots

(a) State the formula for determining the angle between the intersection of two lines.

(b) Show that the angle bisector line: y = Mx between the lines y = m1*x and y = m2*x is given by (M - m1)(1 + M*m2) = (m2 - M)(1 + M*m1).

(c) Hence show that the gradients of the two angle bisector lines is given by the quadratic in M:

(m1 + m2)M^2 + 2(1-m1*m2)M - (m1 + m2) = 0.

(d) State the product of the roots of this equation. Hence deduce the relationship between these two angle bisectors.

N.B. the 1 and 2 in m1 and m2 are the bases of m (like in logs).

2. ## hi

Hi!

I only did the first, maybe this will give you a hint how to proceed.
Let the angle between the positive x-axis and y=m1x be called $\theta$
and let the angle between y=mx and the positive x-axis be called $\alpha$

Then the angle $\beta$ is given by $\theta - \alpha$

How do you figure out $\theta \, \mbox{and } \alpha$ ?

It has something to do with Tangens.
Let´s say you have the line $y = x$
Then $\frac{\Delta y}{\Delta x} = \mbox{ the angle between the positive x-axis and the line}$ where the index 1 and 2 stand for the different lines.
Thus $tan(\frac{\Delta y}{\Delta x}) = \theta$
And $\theta = arctan(\frac{\Delta y}{\Delta x})$

Then you do the same for the other line.
And you end up with:
$\beta = arctan(\frac{\Delta y_{2}}{\Delta x_{2}}) - arctan(\frac{\Delta y_{1}}{\Delta x_{1}})$

3. I can't seem to be able to solve (b), (c) or (d). I have no clue how the equations given can be used to find the stuff. Could someone please give me a hint???