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Let L1 be a straight 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 equation can represent L1 ?
- 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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Let L1 be a straight line passing through the origin and L2 be the st...
Given Information:
- L1 is a straight line passing through the origin.
- L2 is the straight line x + y = 1.
- The circle x^2 + y^2 - x - 3y = 0 intercepts L1 and L2 at the same points.
To find:
Which equation can represent L1?
Solution:
Let's first find the intercepts made by the circle on L1 and L2.
Intercepts on L1:
Since L1 passes through the origin, the equation of L1 can be written as y = mx, where m is the slope of the line.
To find the intercepts made by the circle on L1, we substitute y = mx in the equation of the circle and solve for x:
x^2 + (mx)^2 - x - 3(mx) = 0
x^2 + m^2x^2 - x - 3mx = 0
Combining like terms, we get:
(1 + m^2)x^2 - (1 + 3m)x = 0
For this equation to represent the intercepts made by the circle on L1, the discriminant of this quadratic equation must be zero.
Discriminant = (1 + 3m)^2 - 4(1 + m^2)(0) = (1 + 3m)^2
Setting the discriminant to zero, we have:
(1 + 3m)^2 = 0
Solving this equation, we get:
1 + 3m = 0
3m = -1
m = -1/3
Therefore, the equation of L1 is y = (-1/3)x.
Intercepts on L2:
The equation of L2 is given as x + y = 1.
To find the intercepts made by the circle on L2, we substitute y = 1 - x in the equation of the circle and solve for x:
x^2 + (1 - x)^2 - x - 3(1 - x) = 0
x^2 + 1 - 2x + x^2 - x - 3 + 3x = 0
2x^2 + 2 = 0
x^2 + 1 = 0
This quadratic equation has no real roots, which means the circle does not intercept L2.
Conclusion:
The circle does not intercept L2, so the equation x - y = 0 cannot represent L1.
Therefore, the equation that can represent L1 is y = (-1/3)x, which is option B. Additionally, the equation x + 7y = 0 can also represent L1 as it is a multiple of y = (-1/3)x, giving the same line. So option C is also correct.
- L1 is a straight line passing through the origin.
- L2 is the straight line x + y = 1.
- The circle x^2 + y^2 - x - 3y = 0 intercepts L1 and L2 at the same points.
To find:
Which equation can represent L1?
Solution:
Let's first find the intercepts made by the circle on L1 and L2.
Intercepts on L1:
Since L1 passes through the origin, the equation of L1 can be written as y = mx, where m is the slope of the line.
To find the intercepts made by the circle on L1, we substitute y = mx in the equation of the circle and solve for x:
x^2 + (mx)^2 - x - 3(mx) = 0
x^2 + m^2x^2 - x - 3mx = 0
Combining like terms, we get:
(1 + m^2)x^2 - (1 + 3m)x = 0
For this equation to represent the intercepts made by the circle on L1, the discriminant of this quadratic equation must be zero.
Discriminant = (1 + 3m)^2 - 4(1 + m^2)(0) = (1 + 3m)^2
Setting the discriminant to zero, we have:
(1 + 3m)^2 = 0
Solving this equation, we get:
1 + 3m = 0
3m = -1
m = -1/3
Therefore, the equation of L1 is y = (-1/3)x.
Intercepts on L2:
The equation of L2 is given as x + y = 1.
To find the intercepts made by the circle on L2, we substitute y = 1 - x in the equation of the circle and solve for x:
x^2 + (1 - x)^2 - x - 3(1 - x) = 0
x^2 + 1 - 2x + x^2 - x - 3 + 3x = 0
2x^2 + 2 = 0
x^2 + 1 = 0
This quadratic equation has no real roots, which means the circle does not intercept L2.
Conclusion:
The circle does not intercept L2, so the equation x - y = 0 cannot represent L1.
Therefore, the equation that can represent L1 is y = (-1/3)x, which is option B. Additionally, the equation x + 7y = 0 can also represent L1 as it is a multiple of y = (-1/3)x, giving the same line. So option C is also correct.
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Community Answer
Let L1 be a straight line passing through the origin and L2 be the st...
Let equation of line L1 be y = mx. Intercepts made by L1 and L2 on the circle will be equal ie, L1 and L2 are at the same distance from the centre of the circle.
Centre of the given circle is (1/2,−3/2).
Therefore,
Thus, two chords are x + 7y = 0 and y − x = 0.
Therefore, x − y = 0 and x + 7y = 0 are correct answers.
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Let L1 be a straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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 straight 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 equation can represent L1 ?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
ample number of questions to practice Let L1 be a straight 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 equation can represent L1 ?a)x + y = 0b)x - y = 0c)x + 7y = 0d)x - 7y = 0Correct answer is option 'B,C'. Can you explain this answer? tests, examples and also practice JEE tests.
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