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L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.
The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates is
Correct answer is '47'. Can you explain this answer?
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L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is s...
2x + y = 50
y = mx + 1
Using the value of y = mx + 1,
2x + mx + 1 = 50
x(2 + m) = 49
So, x coordinate of intersection point of
For x to be an integer, 2 + m should be divisible by 49, so m can be 47, 5, -1, or -51.
So, mgreatest = 47 and mleast = -51
Greatest value of m = 47
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L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is s...
Understanding the Lines
The equations of the lines given are:
- L1: 2x + y = 50
- L2: y = mx + 1
To find the point of intersection, we will substitute the equation of L2 into L1.
Substituting L2 into L1
By substituting y from L2 into L1, we get:
- 2x + (mx + 1) = 50
- Rearranging gives: (2 + m)x + 1 = 50
- Thus, (2 + m)x = 49
- Therefore, x = 49 / (2 + m)
Now, substituting x back into L2 for y:
- y = m(49 / (2 + m)) + 1
- Simplifying gives: y = (49m) / (2 + m) + 1
Integral Points Condition
For (x, y) to be integral points, both x and y must be integers.
- For x to be an integer, (2 + m) must divide 49.
- The divisors of 49 are ±1, ±7, ±49.
Calculating m Values
From the divisors, we can find corresponding values for m:
- For divisor 1: m = -1
- For divisor 7: m = 5
- For divisor 49: m = 47
The maximum value of m when considering positive divisors is 47.
Conclusion
Thus, the greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates is:
- m = 47
This satisfies the conditions set by the problem, making it the correct answer.
The equations of the lines given are:
- L1: 2x + y = 50
- L2: y = mx + 1
To find the point of intersection, we will substitute the equation of L2 into L1.
Substituting L2 into L1
By substituting y from L2 into L1, we get:
- 2x + (mx + 1) = 50
- Rearranging gives: (2 + m)x + 1 = 50
- Thus, (2 + m)x = 49
- Therefore, x = 49 / (2 + m)
Now, substituting x back into L2 for y:
- y = m(49 / (2 + m)) + 1
- Simplifying gives: y = (49m) / (2 + m) + 1
Integral Points Condition
For (x, y) to be integral points, both x and y must be integers.
- For x to be an integer, (2 + m) must divide 49.
- The divisors of 49 are ±1, ±7, ±49.
Calculating m Values
From the divisors, we can find corresponding values for m:
- For divisor 1: m = -1
- For divisor 7: m = 5
- For divisor 49: m = 47
The maximum value of m when considering positive divisors is 47.
Conclusion
Thus, the greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates is:
- m = 47
This satisfies the conditions set by the problem, making it the correct answer.
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L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. Can you explain this answer?
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L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. 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 L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. Can you explain this answer?.
L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. 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 L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for L1: 2x + y = 50 and L2: y = mx + 1 are two lines. A point (x, y) is said to be integral point if x, y ∈ I.The greatest integral value of m for which the point of intersection of L1 and L2 has integral coordinates isCorrect answer is '47'. Can you explain this answer?.
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