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The foot of the perpendicular drawn from the origin on the line 3x + y = λ (λ ≠ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:
- a)1 : 9
- b)1 : 3
- c)9 : 1
- d)3 : 1
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The foot of the perpendicular drawn from the origin on the line 3x + y...
Let BP : PA = K : 1
Since slope mOP x slope mAB = -1;
So, PB : PA = K : 1 = 9 : 1
Most Upvoted Answer
The foot of the perpendicular drawn from the origin on the line 3x + y...
Given:
The equation of the line is 3x + y = λ (λ ≠ 0)
P is the foot of the perpendicular drawn from the origin to the line
The line intersects the x-axis at A and the y-axis at B
To Find:
The ratio BP : PA
Solution:
Finding the Coordinates of A, B, and P:
- To find the coordinates of A, set y = 0 in the equation of the line: 3x = λ
- So, x = λ/3, which gives the coordinates of A as (λ/3, 0)
- To find the coordinates of B, set x = 0 in the equation of the line: y = λ
- So, the coordinates of B are (0, λ)
- The equation of the line perpendicular to 3x + y = λ passing through the origin is y = -3x
- Solving the equations 3x + y = λ and y = -3x, we get the coordinates of P as (λ/10, -3λ/10)
Calculating the Ratio BP : PA:
- Using the distance formula, calculate the lengths BP and PA
- BP = √((0 - λ)^2 + (λ + 3λ/10)^2) = √(λ^2 + (13λ/10)^2) = √(λ^2 + 169λ^2/100) = √(269λ^2/100) = λ√269/10
- PA = √((λ/3 - λ/10)^2 + (0 + 3λ/10)^2) = √((7λ/30)^2 + (3λ/10)^2) = √(49λ^2/900 + 9λ^2/100) = √(49λ^2/900 + 81λ^2/900) = √(130λ^2/900) = λ√13/30
- Therefore, the ratio BP : PA = (λ√269/10) : (λ√13/30) = √269/10 : √13/30 = 3√269 : √13
- Simplifying further, we get the ratio as 9 : 1
Therefore, the ratio BP : PA is 9 : 1.
The equation of the line is 3x + y = λ (λ ≠ 0)
P is the foot of the perpendicular drawn from the origin to the line
The line intersects the x-axis at A and the y-axis at B
To Find:
The ratio BP : PA
Solution:
Finding the Coordinates of A, B, and P:
- To find the coordinates of A, set y = 0 in the equation of the line: 3x = λ
- So, x = λ/3, which gives the coordinates of A as (λ/3, 0)
- To find the coordinates of B, set x = 0 in the equation of the line: y = λ
- So, the coordinates of B are (0, λ)
- The equation of the line perpendicular to 3x + y = λ passing through the origin is y = -3x
- Solving the equations 3x + y = λ and y = -3x, we get the coordinates of P as (λ/10, -3λ/10)
Calculating the Ratio BP : PA:
- Using the distance formula, calculate the lengths BP and PA
- BP = √((0 - λ)^2 + (λ + 3λ/10)^2) = √(λ^2 + (13λ/10)^2) = √(λ^2 + 169λ^2/100) = √(269λ^2/100) = λ√269/10
- PA = √((λ/3 - λ/10)^2 + (0 + 3λ/10)^2) = √((7λ/30)^2 + (3λ/10)^2) = √(49λ^2/900 + 9λ^2/100) = √(49λ^2/900 + 81λ^2/900) = √(130λ^2/900) = λ√13/30
- Therefore, the ratio BP : PA = (λ√269/10) : (λ√13/30) = √269/10 : √13/30 = 3√269 : √13
- Simplifying further, we get the ratio as 9 : 1
Therefore, the ratio BP : PA is 9 : 1.
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The foot of the perpendicular drawn from the origin on the line 3x + y =λ(λ≠0)is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:a)1 : 9b)1 : 3c)9 : 1d)3 : 1Correct answer is option 'C'. Can you explain this answer?
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The foot of the perpendicular drawn from the origin on the line 3x + y =λ(λ≠0)is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:a)1 : 9b)1 : 3c)9 : 1d)3 : 1Correct answer is option 'C'. 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 The foot of the perpendicular drawn from the origin on the line 3x + y =λ(λ≠0)is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:a)1 : 9b)1 : 3c)9 : 1d)3 : 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The foot of the perpendicular drawn from the origin on the line 3x + y =λ(λ≠0)is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is:a)1 : 9b)1 : 3c)9 : 1d)3 : 1Correct answer is option 'C'. Can you explain this answer?.
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