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If the algebraic sum of length of perpendicular drawn from (2,0), (0,2), (1,1) to given line is zero then the given line passes through fixed point p. Find its it's coordinates.
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If the algebraic sum of length of perpendicular drawn from (2,0), (0,2...
(1,1) Is answer , it's simple assume line to be ax+by+c= 0 , satisfy the given conditions #perpendicular_distance it will come out as a+b+c=0 , which will follow by 1,1your welcome, do follow ty
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If the algebraic sum of length of perpendicular drawn from (2,0), (0,2...
Understanding the Problem
The problem states that the algebraic sum of the lengths of the perpendiculars drawn from points (2, 0), (0, 2), and (1, 1) to a given line is zero. This implies that the line must pass through a fixed point.
Key Concepts
- The equation of a line can be represented as:
\( ax + by + c = 0 \)
- The formula for the length of the perpendicular from a point (x₁, y₁) to the line ax + by + c = 0 is given by:
\( \text{Distance} = \frac{|ax₁ + by₁ + c|}{\sqrt{a^2 + b^2}} \)
Setting Up the Equation
- For points (2, 0), (0, 2), and (1, 1), we can establish the following equations based on distances:
- From (2, 0):
\( \frac{|2a + 0b + c|}{\sqrt{a^2 + b^2}} \)
- From (0, 2):
\( \frac{|0a + 2b + c|}{\sqrt{a^2 + b^2}} \)
- From (1, 1):
\( \frac{|1a + 1b + c|}{\sqrt{a^2 + b^2}} \)
Sum of Perpendicular Distances
- The algebraic sum of these distances must equal zero:
\( |2a + c| + |2b + c| + |a + b + c| = 0 \)
- This condition implies that all three expressions need to equal zero simultaneously.
Finding the Fixed Point
- Solving the equations gives us the relationships:
\( 2a + c = 0 \)
\( 2b + c = 0 \)
\( a + b + c = 0 \)
- From these relationships, we find:
\( c = -2a \)
\( c = -2b \)
\( a + b - 2a = 0 \Rightarrow b = a \)
- Thus, the line passes through the point:
\( (1, 1) \)
Conclusion
The fixed point through which the line passes is:
(1, 1)
The problem states that the algebraic sum of the lengths of the perpendiculars drawn from points (2, 0), (0, 2), and (1, 1) to a given line is zero. This implies that the line must pass through a fixed point.
Key Concepts
- The equation of a line can be represented as:
\( ax + by + c = 0 \)
- The formula for the length of the perpendicular from a point (x₁, y₁) to the line ax + by + c = 0 is given by:
\( \text{Distance} = \frac{|ax₁ + by₁ + c|}{\sqrt{a^2 + b^2}} \)
Setting Up the Equation
- For points (2, 0), (0, 2), and (1, 1), we can establish the following equations based on distances:
- From (2, 0):
\( \frac{|2a + 0b + c|}{\sqrt{a^2 + b^2}} \)
- From (0, 2):
\( \frac{|0a + 2b + c|}{\sqrt{a^2 + b^2}} \)
- From (1, 1):
\( \frac{|1a + 1b + c|}{\sqrt{a^2 + b^2}} \)
Sum of Perpendicular Distances
- The algebraic sum of these distances must equal zero:
\( |2a + c| + |2b + c| + |a + b + c| = 0 \)
- This condition implies that all three expressions need to equal zero simultaneously.
Finding the Fixed Point
- Solving the equations gives us the relationships:
\( 2a + c = 0 \)
\( 2b + c = 0 \)
\( a + b + c = 0 \)
- From these relationships, we find:
\( c = -2a \)
\( c = -2b \)
\( a + b - 2a = 0 \Rightarrow b = a \)
- Thus, the line passes through the point:
\( (1, 1) \)
Conclusion
The fixed point through which the line passes is:
(1, 1)
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If the algebraic sum of length of perpendicular drawn from (2,0), (0,2), (1,1) to given line is zero then the given line passes through fixed point p. Find its it's coordinates.
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If the algebraic sum of length of perpendicular drawn from (2,0), (0,2), (1,1) to given line is zero then the given line passes through fixed point p. Find its it's coordinates. for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the algebraic sum of length of perpendicular drawn from (2,0), (0,2), (1,1) to given line is zero then the given line passes through fixed point p. Find its it's coordinates. covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the algebraic sum of length of perpendicular drawn from (2,0), (0,2), (1,1) to given line is zero then the given line passes through fixed point p. Find its it's coordinates..
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