Find an equation of the line that bisects the obtuse angles formed by the lines with equations 3x-y=1 and x+y=-2.

a. (3√2 +√10)x-(√10 + √2)y-2√10+√2=0
b. (3√2 - √10)x+(√10 - √2)y+2√10+√2=0
c. (3√2 + √10)x+(√10 - √2)y+2√10 - √2=0
d. (3√2 + √10)x-(√10 + √2)y-2√10-√2=0
i'm not sure on this one!

before I answer this question, I have to establish if you know

1. that slope = tan(angle the line makes with the x-axis)
2. if m1 and m2 are the slopes of 2 nonparallel lines, and theta is the acute angle between them, then
tan(theta) = │(m1 - m2)/(1 + m1m2)│

If not please watch this short video.

http://www.slideshare.net/nsimmons/11-x1-t05-07-angle-between-two-lines

3. tan(2A) = 2tanA/(1-tan^2 A)

I also noticed that in the first answer given, the slope of the line is
(3√2+√10)/(√10+√2) which when rationalized is (1+√5)/2 which is the answer that I got for the required line.

So you might have obtained the correct slope of the new line without realizing it.

Let me know if you need more help.

okay thanks

To find the equation of the line that bisects the obtuse angles formed by the given lines, we need to follow these steps:

1. Find the angles between the given lines.
2. Determine the slopes of the given lines.
3. Calculate the average of the slopes of the given lines.
4. Find the perpendicular slope to the average slope.
5. Use the average slope and the perpendicular slope to determine a point on the bisecting line.
6. Use the point found in step 5 to obtain the equation of the line.

Let's go through each step:

Step 1: Finding the angles between the lines
First, we need to find the angle between the lines. We know that the tangent of the angle between two lines is given by the formula:

tan(θ) = |(m1 - m2) / (1 + m1 * m2)|

where m1 and m2 are the slopes of the lines.

For the given lines:
Line 1: 3x - y = 1
Line 2: x + y = -2

Converting these equations into slope-intercept form (y = mx + b), we get:
Line 1: y = 3x - 1
Line 2: y = -x - 2

The slopes of the lines are:
m1 = 3
m2 = -1

Using the formula mentioned earlier, we can calculate the tangent of the angle between the lines:

tan(θ) = |(3 - (-1)) / (1 + 3 * (-1))|
= |4 / 2|
= 2

Taking the inverse tangent (arctan) of 2 gives us the angle between the lines:

θ = arctan(2)

Using a calculator, we find:

θ ≈ 63.43 degrees

Since we are looking for the bisecting line of the obtuse angle, which is greater than 90 degrees, we can conclude that the angle between the lines is obtuse.

Step 2: Determine the slopes of the given lines
From the equations in slope-intercept form, we already found the slopes of the given lines:
m1 = 3
m2 = -1

Step 3: Calculate the average of the slopes
To find the average slope, we need to add the slopes and divide by 2:
Average slope = (m1 + m2) / 2
= (3 + (-1)) / 2
= 2 / 2
= 1

Step 4: Find the perpendicular slope to the average slope
The slopes of perpendicular lines are negative reciprocals of each other. So, we need to find the negative reciprocal of the average slope:
Perpendicular slope = -1 / Average slope
= -1 / 1
= -1

Step 5: Use the average slope and the perpendicular slope to determine a point
To find a point on the bisecting line, we can use the equations of the given lines and solve them simultaneously.

The given equations are:
Line 1: 3x - y = 1
Line 2: x + y = -2

We can solve them using elimination or substitution.

Using substitution, we solve Line 2 for x:
x = -y - 2

Substituting this value of x into Line 1, we get:
3(-y - 2) - y = 1
-3y - 6 - y = 1
-4y - 6 = 1
-4y = 7
y = -7/4

Substituting this value of y back into Line 2, we can find x:
x + (-7/4) = -2
x = -2 + 7/4
x = -8/4 + 7/4
x = -1/4

So, the point on the bisecting line is (-1/4, -7/4).

Step 6: Use the point to obtain the equation of the line
Now that we have the slope of the bisecting line as -1 and a point on that line as (-1/4, -7/4), we can use the point-slope form of a line to find the equation.

Using the point-slope form:
y - y1 = m(x - x1)

Substituting the values, we get:
y - (-7/4) = -1(x - (-1/4))
y + 7/4 = -(x + 1/4)

Multiplying through by 4 to get rid of the fractions:
4y + 7 = -4(x + 1/4)

Expanding and rearranging:
4y + 7 = -4x - 1
4y + 4x = -1 - 7
4y + 4x = -8

Dividing throughout by 4:
y + x = -2

Rearranging to the standard form:
x + y + 2 = 0

So, the equation of the line that bisects the obtuse angles formed by the given lines is option:

a. (3√2 +√10)x - (√10 + √2)y - 2√10 + √2 = 0

Therefore, the correct answer is (a).