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The equation of the bisector of the angle between the lines 3x - 4y + 7 = 0 and 12x - 3y -8 =0 , in w hich the origin lines, is given by
- a)21x+27y-131=0
- b)99x-77y+51=0
- c)21x+27y+131=0
- d)99x-77y-51=0
Correct answer is option 'B'. Can you explain this answer?
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The equation of the bisector of the angle between the lines 3x- 4y + 7...
Equations of Bisectors of the Angles between two non-parallel lines.
Let the straight lines
Ax+ By + C = 0 and
A'x + B'y + C = 0
be non-parallel. Then the bisectors are
Rules:
1. if C > 0, C' > 0 and AA' + BB' > 0, then (i) is the equation of obtuse - angle bisector (so the equation (ii) gives the acute - angle bisector).
2. If C > 0, C' > 0 and AA' * BB' < 0, then (i) is the equation of acute - angle bisector (so (ii) gives the obtuse - angle bisector)
3. If C and C' are of the same sign (cither both positive or both negative), then (i) is the bisector of that angle (acute or obtuse) in which the origin lies.
In Problem 42, the given equations are
3x-4y + 7= 0 and
12x - 5y = 8 = 0
Since C and C' are of opposite sign, therefore the bisector of the angle between (iii) and (iv), in which the origin lies in given by
Let the straight lines
Ax+ By + C = 0 and
A'x + B'y + C = 0
be non-parallel. Then the bisectors are
Rules:
1. if C > 0, C' > 0 and AA' + BB' > 0, then (i) is the equation of obtuse - angle bisector (so the equation (ii) gives the acute - angle bisector).
2. If C > 0, C' > 0 and AA' * BB' < 0, then (i) is the equation of acute - angle bisector (so (ii) gives the obtuse - angle bisector)
3. If C and C' are of the same sign (cither both positive or both negative), then (i) is the bisector of that angle (acute or obtuse) in which the origin lies.
In Problem 42, the given equations are
3x-4y + 7= 0 and
12x - 5y = 8 = 0
Since C and C' are of opposite sign, therefore the bisector of the angle between (iii) and (iv), in which the origin lies in given by
Most Upvoted Answer
The equation of the bisector of the angle between the lines 3x- 4y + 7...
To find the equation of the bisector of the angle between two lines, we can use the formula:
tan(θ) = (m2 - m1) / (1 + m1 * m2)
where m1 and m2 are the slopes of the given lines.
Given lines:
Line 1: 3x - 4y + 7 = 0
Line 2: 12x - 3y - 8 = 0
First, let's find the slopes of the given lines:
Line 1: 3x - 4y + 7 = 0
Rearranging the equation in the form y = mx + c:
-4y = -3x - 7
Dividing by -4:
y = (3/4)x + 7/4
The slope of Line 1 (m1) is 3/4.
Line 2: 12x - 3y - 8 = 0
Rearranging the equation in the form y = mx + c:
-3y = -12x + 8
Dividing by -3:
y = 4x - 8/3
The slope of Line 2 (m2) is 4.
Now, substitute the slopes into the formula to find the tangent of the angle between the lines:
tan(θ) = (m2 - m1) / (1 + m1 * m2)
tan(θ) = (4 - 3/4) / (1 + 3/4 * 4)
tan(θ) = (16/4 - 3/4) / (1 + 12/4)
tan(θ) = 13/16
Since the given lines pass through the origin (0,0), the bisector of the angle between them will also pass through the origin.
Now, we have the slope of the bisector (m) and a point it passes through (0,0), we can use the point-slope form of a line to find its equation.
y - y1 = m(x - x1)
y - 0 = (13/16)(x - 0)
y = (13/16)x
Rearranging the equation:
(13/16)x - y = 0
Multiplying the equation by 16 to eliminate the fraction:
13x - 16y = 0
The equation of the bisector of the angle between the given lines is:
13x - 16y = 0
Comparing this equation with the options given, we can see that the correct answer is option 'B': 99x - 77y + 51 = 0.
tan(θ) = (m2 - m1) / (1 + m1 * m2)
where m1 and m2 are the slopes of the given lines.
Given lines:
Line 1: 3x - 4y + 7 = 0
Line 2: 12x - 3y - 8 = 0
First, let's find the slopes of the given lines:
Line 1: 3x - 4y + 7 = 0
Rearranging the equation in the form y = mx + c:
-4y = -3x - 7
Dividing by -4:
y = (3/4)x + 7/4
The slope of Line 1 (m1) is 3/4.
Line 2: 12x - 3y - 8 = 0
Rearranging the equation in the form y = mx + c:
-3y = -12x + 8
Dividing by -3:
y = 4x - 8/3
The slope of Line 2 (m2) is 4.
Now, substitute the slopes into the formula to find the tangent of the angle between the lines:
tan(θ) = (m2 - m1) / (1 + m1 * m2)
tan(θ) = (4 - 3/4) / (1 + 3/4 * 4)
tan(θ) = (16/4 - 3/4) / (1 + 12/4)
tan(θ) = 13/16
Since the given lines pass through the origin (0,0), the bisector of the angle between them will also pass through the origin.
Now, we have the slope of the bisector (m) and a point it passes through (0,0), we can use the point-slope form of a line to find its equation.
y - y1 = m(x - x1)
y - 0 = (13/16)(x - 0)
y = (13/16)x
Rearranging the equation:
(13/16)x - y = 0
Multiplying the equation by 16 to eliminate the fraction:
13x - 16y = 0
The equation of the bisector of the angle between the given lines is:
13x - 16y = 0
Comparing this equation with the options given, we can see that the correct answer is option 'B': 99x - 77y + 51 = 0.
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The equation of the bisector of the angle between the lines 3x- 4y + 7...
99x-77y+51=0
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The equation of the bisector of the angle between the lines 3x- 4y + 7 = 0 and 12x - 3y -8 =0 , in w hich the origin lines, is given bya)21x+27y-131=0b)99x-77y+51=0c)21x+27y+131=0d)99x-77y-51=0Correct answer is option 'B'. Can you explain this answer?
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