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Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)
- a)x + y = 0
- b)x – y = 0
- c)x + 7y = 0
- d)x – 7y = 0
Correct answer is option 'B,C'. Can you explain this answer?
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Verified Answer
Let L1 be a strainght line passing through the origin and L2 be the st...
We know that length of intercept made by a circle on a line is given by = 
where p = ⊥ distance of line from the centre of the circle.
Here circle is x2 + y2 – x + 3y = 0 with centre 
and radius = 
L1 : y = mx (any line through origin)
L2 : x + y – 1 = 0 (given line)
ATQ circle makes equal intercepts on L1 and L2
⇒ m2 + 6m + 9 = 8m2 + 8 ⇒ 7m2 – 6m – 1 = 0
⇒ 7m2 – 7m + m – 1 = 0 ⇒ (7m + 1) (m – 1) = 0
⇒ m = 1, – 1/7
∴ The required line L1 is y = x or y =
i.e., x – y = 0 or x + 7y = 0.
Most Upvoted Answer
Let L1 be a strainght line passing through the origin and L2 be the st...
Understanding the Problem
To find the equations representing the line L1 that passes through the origin and has equal intercepts with the circle defined by the equation x² + y² - x + 3y = 0 and the line L2 (x + y = 1), we need to analyze both lines and the circle.
Circle Equation Analysis
1. Rewrite the circle equation:
- Rearranging gives us: (x - 0.5)² + (y + 1.5)² = 2.25, indicating a circle centered at (0.5, -1.5) with a radius of 1.5.
Intercepts on L2 (x + y = 1)
1. Find intercepts:
- When x = 0, y = 1 (point (0, 1)).
- When y = 0, x = 1 (point (1, 0)).
- Length of intercept = sqrt(1² + 1²) = sqrt(2).
Finding L1's Equation
1. General form:
- L1 can be represented as y = mx, where m is the slope.
2. Intercepts on L1:
- The x-intercept is (a, 0) and y-intercept is (0, b).
- The intercept lengths are equal when the line passes through the origin.
Setting the Conditions for Equal Intercepts
1. For L1 and L2 to have equal intercepts:
- The distance from the origin to the intercepts on L1 must equal sqrt(2) (the length of the intercept on L2).
- This implies relationships between the slope m and the intercept lengths.
Testing the Given Options
1. Option (a) x + y = 0:
- Slope = -1 (intercepts would not be equal).
2. Option (b) x - y = 0:
- Slope = 1 (equal intercepts).
3. Option (c) x + 7y = 0:
- Slope = -1/7 (equal intercepts).
4. Option (d) x - 7y = 0:
- Slope = 7 (not equal).
Conclusion
The suitable equations for L1 that maintain equal intercepts with L2 are option (b) x - y = 0 and option (c) x + 7y = 0.
To find the equations representing the line L1 that passes through the origin and has equal intercepts with the circle defined by the equation x² + y² - x + 3y = 0 and the line L2 (x + y = 1), we need to analyze both lines and the circle.
Circle Equation Analysis
1. Rewrite the circle equation:
- Rearranging gives us: (x - 0.5)² + (y + 1.5)² = 2.25, indicating a circle centered at (0.5, -1.5) with a radius of 1.5.
Intercepts on L2 (x + y = 1)
1. Find intercepts:
- When x = 0, y = 1 (point (0, 1)).
- When y = 0, x = 1 (point (1, 0)).
- Length of intercept = sqrt(1² + 1²) = sqrt(2).
Finding L1's Equation
1. General form:
- L1 can be represented as y = mx, where m is the slope.
2. Intercepts on L1:
- The x-intercept is (a, 0) and y-intercept is (0, b).
- The intercept lengths are equal when the line passes through the origin.
Setting the Conditions for Equal Intercepts
1. For L1 and L2 to have equal intercepts:
- The distance from the origin to the intercepts on L1 must equal sqrt(2) (the length of the intercept on L2).
- This implies relationships between the slope m and the intercept lengths.
Testing the Given Options
1. Option (a) x + y = 0:
- Slope = -1 (intercepts would not be equal).
2. Option (b) x - y = 0:
- Slope = 1 (equal intercepts).
3. Option (c) x + 7y = 0:
- Slope = -1/7 (equal intercepts).
4. Option (d) x - 7y = 0:
- Slope = 7 (not equal).
Conclusion
The suitable equations for L1 that maintain equal intercepts with L2 are option (b) x - y = 0 and option (c) x + 7y = 0.
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Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer?
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Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer?.
Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer?.
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Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer?, a detailed solution for Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? has been provided alongside types of Let L1 be a strainght line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle x2 + y2 - x + 3y= 0 on L1 and L2 are equal, then which of the following equations can represent L1? (1999 - 3 Marks)a)x + y = 0b)x – y = 0c)x + 7y = 0d)x – 7y = 0Correct answer is option 'B,C'. Can you explain this answer? theory, EduRev gives you an
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