How do find if (1,2) lies in between the acute or obtuse region of two lines .

Question

The two lines are $\sqrt{3}x-y+5=0$ and $\sqrt{3}x+y-1=0$.

How will I find in which region the point lies: the acute region or the obtuse region?

I feel this can't be done if anyone could suggest a method that would be great

Answer 1

It can be done, and quite easily!

Now for any point $P(x_1, y_1)$ , if I put it into the formula of the line $L: ax+by+c$, I get three outcomes

1. $ax_1 + by_1 + c < 0$ - the point is in the lower half space created by the line

2. $ax_1 + by_1 + c = 0$ - the point is on the line

3. $ax_1 + by_1 + c > 0$ - the point is in the upper half space created by the line

Now the primary angle between the two lines is $60^0$ (based on comparing slopes)

Therefore any point which satisfies the below is in the "obtuse" portion

$$(\sqrt3 x-y+5)(\sqrt3 +y-1) > 0$$