Find the bisector of the obtuse angle between the lines 11x +2y + 7 = 0 and x + 2y + 20 = 0
Well, aren't we ob-tuse with these obtuse angles? So, to find the bisector of the obtuse angle between those lines, we need to find the slope of each line first.
Line 1: 11x + 2y + 7 = 0
Let's put that in slope-intercept form:
2y = -11x - 7
y = (-11/2)x - 7/2
Line 2: x + 2y + 20 = 0
In slope-intercept form:
2y = -x - 20
y = (-1/2)x - 10
Now we have the slopes: -11/2 and -1/2.
The bisector of an angle is always perpendicular to both the given lines. So, let's find the perpendicular slope by flipping the fractions and changing the sign.
For Line 1, the perpendicular slope will be: 2/11 (flipped and sign changed).
For Line 2, the perpendicular slope will be: 2 (flipped and sign changed).
Now, we have two lines with their perpendicular slopes. We can find the intersection point of these lines and that will be the bisector of the obtuse angle.
Using the point-slope form with the intersection point (let's call it (x, y)):
y - y1 = m(x - x1)
For Line 1 (perpendicular slope 2/11):
y - y1 = (2/11)(x - x1)
Substituting the coordinates of the intersection point:
y - y1 = (2/11)(x - x1)
For Line 2 (perpendicular slope 2):
y - y1 = 2(x - x1)
Substituting the coordinates of the intersection point:
y - y1 = 2(x - x1)
We have two equations now. Solve them to find the intersection point and voila! You've got yourself the bisector of the obtuse angle.