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Math Chapter 8 : Topic 2
Family of Lines
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The family of lines passing through the intersection of two lines is represented by:
Reasoning: This is the standard form of a family of lines through intersection.
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In $L_{1} + \lambda\ L_{2} = 0$, the parameter $\lambda$ represents:
Reasoning: Different values of $\lambda$ give different lines.
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The family of lines passes through:
Reasoning: It is constructed from two intersecting lines.
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If $L_{1} = 0$ and $L_{2} = 0$, then their combination represents:
Reasoning: Linear combination forms family of lines.
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For two lines $L_{1} = 0$, $L_{2} = 0$, their intersection point satisfies:
Reasoning: Intersection satisfies both lines.
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Which value of $\lambda$ gives a specific line in the family?
Reasoning: Each real value gives a different line.
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The family of lines is also known as:
Reasoning: All lines pass through a common point.
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If $\lambda = 0$, equation becomes:
Reasoning: Only first line remains.
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If $\lambda = - \ 1$, equation becomes:
Reasoning: Substituting gives sum of equations.
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The concept is used to find:
Reasoning: Generates infinite lines through intersection.
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If two lines are parallel, their family:
Reasoning: But no unique intersection point.
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The expression $L_{1} + \lambda\ L_{2} = 0$ is:
Reasoning: Combination of linear equations.
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Which geometric property is essential?
Reasoning: Family depends on common point.
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If two lines are identical, family becomes:
Reasoning: Both equations same line.
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Family of lines helps in:
Reasoning: Used in advanced line analysis.
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Which operation generates the family?
Reasoning: Linear combination creates family.
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If $L_{1} = x + y - 1$, $L_{2} = x - y$, then family passes through:
Reasoning: Both lines define common point.
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The family always contains:
Reasoning: $\lambda$ varies continuously.
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Which type of lines form this family?
Reasoning: Must intersect at a point.
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Changing $\lambda$ affects:
Reasoning: Each $\lambda$ gives different line.
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If $L_{1} + \lambda\ L_{2} = 0$, then at intersection point:
Reasoning: Intersection satisfies both equations.
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Which is NOT required for family of lines?
Reasoning: Distance is unrelated.
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Family of lines concept is extension of:
Reasoning: Based on linear algebra.
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If $\lambda$ is fixed, equation represents:
Reasoning: Specific $\lambda$ gives one line.
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The key idea of family of lines is:
Reasoning: All lines pass through same point.
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