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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?
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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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Community 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
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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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