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Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2 + MC2 is minimum is ________. (in integer)
Correct answer is '5'. Can you explain this answer?
Verified Answer
Let AD and BC be two vertical poles at A and B, respectively, on a hor...
(MD)2 + (MC)2 = h2 + 64 + (h - 10)2 + 121
= 2h2 - 20h + 64 + 100 + 121
= 2(h2 - 10h) + 285
= 2(h - 5)2 + 235
It is minimum if h = 5.
= 2h2 - 20h + 64 + 100 + 121
= 2(h2 - 10h) + 285
= 2(h - 5)2 + 235
It is minimum if h = 5.
Most Upvoted Answer
Let AD and BC be two vertical poles at A and B, respectively, on a hor...
Understanding the Problem
To find the point M on line segment AB that minimizes the expression MD² + MC², we need to visualize the scenario:
- AD = 8 m (Height of pole at A)
- BC = 11 m (Height of pole at B)
- AB = 10 m (Distance between the two poles)
Setting Up the Coordinates
1. Place point A at (0, 0) and point B at (10, 0) on a Cartesian plane.
2. The coordinates for points D and C can be defined as:
- D (0, 8)
- C (10, 11)
Defining Point M
- Let point M on line segment AB be represented as (x, 0), where 0 ≤ x ≤ 10.
Calculating Distances
1. The distance MD (from M to D) is given by:
- MD = √[(x - 0)² + (0 - 8)²] = √(x² + 64)
2. The distance MC (from M to C) is given by:
- MC = √[(x - 10)² + (0 - 11)²] = √((x - 10)² + 121)
Minimizing the Sum of Distances
- We want to minimize the expression MD² + MC²:
MD² + MC² = (x² + 64) + ((x - 10)² + 121)
- This simplifies to:
x² + 64 + (x² - 20x + 100 + 121)
- Combining terms gives:
2x² - 20x + 285
Finding the Minimum Point
- To find the minimum, use the vertex formula x = -b/(2a):
- Here, a = 2 and b = -20.
- Therefore, x = 20/(2*2) = 5.
Conclusion
The point M that minimizes MD² + MC² is located at a distance of 5 meters from point A. Thus, the required answer is 5.
To find the point M on line segment AB that minimizes the expression MD² + MC², we need to visualize the scenario:
- AD = 8 m (Height of pole at A)
- BC = 11 m (Height of pole at B)
- AB = 10 m (Distance between the two poles)
Setting Up the Coordinates
1. Place point A at (0, 0) and point B at (10, 0) on a Cartesian plane.
2. The coordinates for points D and C can be defined as:
- D (0, 8)
- C (10, 11)
Defining Point M
- Let point M on line segment AB be represented as (x, 0), where 0 ≤ x ≤ 10.
Calculating Distances
1. The distance MD (from M to D) is given by:
- MD = √[(x - 0)² + (0 - 8)²] = √(x² + 64)
2. The distance MC (from M to C) is given by:
- MC = √[(x - 10)² + (0 - 11)²] = √((x - 10)² + 121)
Minimizing the Sum of Distances
- We want to minimize the expression MD² + MC²:
MD² + MC² = (x² + 64) + ((x - 10)² + 121)
- This simplifies to:
x² + 64 + (x² - 20x + 100 + 121)
- Combining terms gives:
2x² - 20x + 285
Finding the Minimum Point
- To find the minimum, use the vertex formula x = -b/(2a):
- Here, a = 2 and b = -20.
- Therefore, x = 20/(2*2) = 5.
Conclusion
The point M that minimizes MD² + MC² is located at a distance of 5 meters from point A. Thus, the required answer is 5.
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Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer?
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Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer?.
Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer?.
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Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer?, a detailed solution for Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? has been provided alongside types of Let AD and BC be two vertical poles at A and B, respectively, on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m, then the distance (in metres) of a point M on AB from the point A such that MD2+ MC2is minimum is ________. (in integer)Correct answer is '5'. Can you explain this answer? theory, EduRev gives you an
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