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Let L(1)=0and L(2) =0 be two intarecting...

Let `L_(1)=0and L_(2) =0` be two intarecting straight lines. Then the number of points, whose distacne from `L_(1)` is 2 units and from `L_(2)` 2 units is

A

one

B

zero

C

four

D

infinite

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the number of points that are at a distance of 2 units from both lines \( L_1 \) and \( L_2 \), which are defined as \( L_1 = 0 \) and \( L_2 = 0 \). ### Step-by-Step Solution: 1. **Identify the Lines**: - The lines \( L_1 \) and \( L_2 \) are both represented by the equation \( y = 0 \) (the x-axis). Since they are intersecting straight lines, we can assume they are the same line for this problem. 2. **Understanding Distance from a Line**: - The distance from a line in the plane can be measured perpendicularly. For a horizontal line like \( L_1 \) (the x-axis), a point at a distance of 2 units can be either 2 units above or 2 units below the line. 3. **Finding Parallel Lines**: - The points that are 2 units away from \( L_1 \) will lie on two parallel lines: - \( y = 2 \) (2 units above \( L_1 \)) - \( y = -2 \) (2 units below \( L_1 \)) 4. **Finding Distance from \( L_2 \)**: - Similarly, since \( L_2 \) is also the x-axis, the points that are 2 units away from \( L_2 \) will also lie on the same two parallel lines: - \( y = 2 \) - \( y = -2 \) 5. **Intersection of Lines**: - Now, we need to find the points that are at a distance of 2 units from both \( L_1 \) and \( L_2 \). This means we are looking for points that lie on both sets of parallel lines: - The lines \( y = 2 \) and \( y = -2 \) are the only lines we need to consider. 6. **Finding Points of Intersection**: - The lines \( y = 2 \) and \( y = -2 \) are horizontal lines. The points on these lines can be represented as: - For \( y = 2 \): All points of the form \( (x, 2) \) where \( x \) can be any real number. - For \( y = -2 \): All points of the form \( (x, -2) \) where \( x \) can be any real number. 7. **Conclusion**: - Since both lines \( y = 2 \) and \( y = -2 \) extend infinitely in the x-direction, there are infinitely many points that satisfy the condition of being 2 units away from both lines \( L_1 \) and \( L_2 \). ### Final Answer: The number of points whose distance from \( L_1 \) is 2 units and from \( L_2 \) is also 2 units is **infinite**. ---
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