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If A,B and C are the angles of a triangle and a,b,c are the corresponding sides, then the equations (sin A + sin B)x+(sin B + sin C)y+sin C + sin A = 0 and (a+b)x + (b+c)y+c+a = 0 have
- a)One solution
- b)No solution
- c)Infinite solution
- d)Three solutions
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If A,B and C are the angles of a triangle and a,b,c are the correspond...
(sin A + sin B)x+(sin B + sin C)y+sin C + sin A = 0 -- Eq (1)
(a+b)x+(b+c)y+c+a = 0 -- Eq (2)
In a triangle, a/sin A = b/sin B = c/sin C = 2R where R is the circumradii
sin A = a/2R, sin B = b/2R, sin C = c/2R
On substituting the values in (sin A + sin B)x+(sin B + sin C)y+sin C + sin A = 0, we get
(a+b)x+(b+c)y+c+a = 0
which is same as Eq 2
Hence the equations will have infinite solutions.
(a+b)x+(b+c)y+c+a = 0 -- Eq (2)
In a triangle, a/sin A = b/sin B = c/sin C = 2R where R is the circumradii
sin A = a/2R, sin B = b/2R, sin C = c/2R
On substituting the values in (sin A + sin B)x+(sin B + sin C)y+sin C + sin A = 0, we get
(a+b)x+(b+c)y+c+a = 0
which is same as Eq 2
Hence the equations will have infinite solutions.
Most Upvoted Answer
If A,B and C are the angles of a triangle and a,b,c are the correspond...
Explanation:
To understand why the equations have infinite solutions, we need to analyze the equations and the properties of triangles.
Equation 1: (sin A sin B)x (sin B sin C)y sin C sin A = 0
Let's break down the equation:
(sin A sin B)x (sin B sin C)y sin C sin A = 0
We know that sin A, sin B, and sin C are all positive values because they are the ratios of the lengths of the sides of the triangle.
Therefore, for the equation to be satisfied, either (sin A sin B) = 0, (sin B sin C) = 0, or sin C sin A = 0.
Now, let's consider each case:
1. (sin A sin B) = 0:
This implies that either sin A = 0 or sin B = 0.
- If sin A = 0, then angle A must be 0 degrees or 180 degrees. In both cases, the triangle would become degenerate, and it would not be a triangle anymore.
- Similarly, if sin B = 0, then angle B must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
2. (sin B sin C) = 0:
This implies that either sin B = 0 or sin C = 0.
- If sin B = 0, then angle B must be 0 degrees or 180 degrees, which would lead to a degenerate triangle.
- Similarly, if sin C = 0, then angle C must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
3. sin C sin A = 0:
This implies that either sin C = 0 or sin A = 0.
- If sin C = 0, then angle C must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
- Similarly, if sin A = 0, then angle A must be 0 degrees or 180 degrees, leading to a degenerate triangle.
Equation 2: (a b)x (b c)y c a = 0
The equation can be written as:
(a b)x (b c)y c a = 0
Expanding the equation, we get:
abx + bcy - ca = 0
This equation represents a linear equation in terms of x and y. It can be rearranged as:
abx - ca = -bcy
Dividing both sides by ab - bc, we get:
x = (-bcy + ca)/(ab - bc)
The equation has a unique solution for x and y if ab - bc ≠ 0. However, if ab - bc = 0, then the equation is satisfied for any value of y, resulting in infinite solutions.
Since the equation can have infinite solutions, the correct answer is option 'C' - Infinite solutions.
To understand why the equations have infinite solutions, we need to analyze the equations and the properties of triangles.
Equation 1: (sin A sin B)x (sin B sin C)y sin C sin A = 0
Let's break down the equation:
(sin A sin B)x (sin B sin C)y sin C sin A = 0
We know that sin A, sin B, and sin C are all positive values because they are the ratios of the lengths of the sides of the triangle.
Therefore, for the equation to be satisfied, either (sin A sin B) = 0, (sin B sin C) = 0, or sin C sin A = 0.
Now, let's consider each case:
1. (sin A sin B) = 0:
This implies that either sin A = 0 or sin B = 0.
- If sin A = 0, then angle A must be 0 degrees or 180 degrees. In both cases, the triangle would become degenerate, and it would not be a triangle anymore.
- Similarly, if sin B = 0, then angle B must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
2. (sin B sin C) = 0:
This implies that either sin B = 0 or sin C = 0.
- If sin B = 0, then angle B must be 0 degrees or 180 degrees, which would lead to a degenerate triangle.
- Similarly, if sin C = 0, then angle C must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
3. sin C sin A = 0:
This implies that either sin C = 0 or sin A = 0.
- If sin C = 0, then angle C must be 0 degrees or 180 degrees, resulting in a degenerate triangle.
- Similarly, if sin A = 0, then angle A must be 0 degrees or 180 degrees, leading to a degenerate triangle.
Equation 2: (a b)x (b c)y c a = 0
The equation can be written as:
(a b)x (b c)y c a = 0
Expanding the equation, we get:
abx + bcy - ca = 0
This equation represents a linear equation in terms of x and y. It can be rearranged as:
abx - ca = -bcy
Dividing both sides by ab - bc, we get:
x = (-bcy + ca)/(ab - bc)
The equation has a unique solution for x and y if ab - bc ≠ 0. However, if ab - bc = 0, then the equation is satisfied for any value of y, resulting in infinite solutions.
Since the equation can have infinite solutions, the correct answer is option 'C' - Infinite solutions.
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If A,B and C are the angles of a triangle and a,b,c are the corresponding sides, then the equations (sin A + sin B)x+(sin B + sin C)y+sin C + sin A = 0 and (a+b)x + (b+c)y+c+a = 0 havea)One solutionb)No solutionc)Infinite solutiond)Three solutionsCorrect answer is option 'C'. Can you explain this answer?
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