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A homogeneous equation of nth degree represents
- a)n straight lines passing through the origin
- b)At the most n straight lines passing through the origin
- c)At least one straight line passing through the origin
- d)n straight lines all of which either pass through origin or may not pass through the origin
Correct answer is option 'B'. Can you explain this answer?
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A homogeneous equation of nth degree representsa)n straight lines pass...
A homogeneous equation of nth degree in x and y is a0xn + a1xn-1 + a2xn-2+ a3xn-3 .....+ an-1xyn-1+ anyn=0 and it represents at most n straight lines passing through the origin.
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A homogeneous equation of nth degree representsa)n straight lines pass...
Understanding Homogeneous Equations of nth Degree
Homogeneous equations of the form \( f(x, y) = 0 \) where \( f \) is a polynomial of degree \( n \) have specific geometric interpretations, especially in the context of lines in the coordinate plane.
Characteristics of Homogeneous Equations:
- **Definition**: A homogeneous equation is one where every term is of the same degree. For instance, \( ax^n + by^n + cxy^{n-1} = 0 \) is a homogeneous equation of degree \( n \).
- **Lines through the Origin**: The solutions to these equations can be interpreted as curves or lines in a two-dimensional space. Since the equation is homogeneous, if \( (x, y) \) is a solution, then \( (kx, ky) \) is also a solution for any non-zero scalar \( k \). This property indicates that the solutions will radiate from the origin.
Reasoning for Option B:
- **At Most n Straight Lines**: A homogeneous equation of degree \( n \) can have at most \( n \) distinct solutions (or lines) that pass through the origin. This arises from the fact that the equation can be factored to yield at most \( n \) linear factors, each representing a line through the origin.
- **Example**: For a quadratic equation \( ax^2 + by^2 + cxy = 0 \), it can be shown that it can represent up to 2 lines through the origin.
Conclusion:
In summary, a homogeneous equation of nth degree represents at most n straight lines passing through the origin due to the nature of polynomial equations and their solutions in the coordinate plane. Thus, the correct answer is option 'B'.
Homogeneous equations of the form \( f(x, y) = 0 \) where \( f \) is a polynomial of degree \( n \) have specific geometric interpretations, especially in the context of lines in the coordinate plane.
Characteristics of Homogeneous Equations:
- **Definition**: A homogeneous equation is one where every term is of the same degree. For instance, \( ax^n + by^n + cxy^{n-1} = 0 \) is a homogeneous equation of degree \( n \).
- **Lines through the Origin**: The solutions to these equations can be interpreted as curves or lines in a two-dimensional space. Since the equation is homogeneous, if \( (x, y) \) is a solution, then \( (kx, ky) \) is also a solution for any non-zero scalar \( k \). This property indicates that the solutions will radiate from the origin.
Reasoning for Option B:
- **At Most n Straight Lines**: A homogeneous equation of degree \( n \) can have at most \( n \) distinct solutions (or lines) that pass through the origin. This arises from the fact that the equation can be factored to yield at most \( n \) linear factors, each representing a line through the origin.
- **Example**: For a quadratic equation \( ax^2 + by^2 + cxy = 0 \), it can be shown that it can represent up to 2 lines through the origin.
Conclusion:
In summary, a homogeneous equation of nth degree represents at most n straight lines passing through the origin due to the nature of polynomial equations and their solutions in the coordinate plane. Thus, the correct answer is option 'B'.
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