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If 2y cos θ = x sin θ and 2x sec θ – y cosec θ = 3, then the relation between x and y is (SSC CGL 2nd Sit. 2012)
- a)2x2 + y2 = 2
- b)x2 + 4y2 = 4
- c)x2 + 4y2 = 1
- d)4x2 + y2 = 4
Correct answer is option 'B'. Can you explain this answer?
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If 2y cos θ = x sin θ and 2x sec θ – y cosec &...
2y cos θ = x sin θ
Most Upvoted Answer
If 2y cos θ = x sin θ and 2x sec θ – y cosec &...
Given Equations
- We have two equations:
1. 2y cos θ = x sin θ
2. 2x sec θ - y cosec θ = 3
Step 1: Manipulate the First Equation
- From the first equation, we can express y in terms of x:
y = (x sin θ) / (2 cos θ)
Step 2: Substitute y in the Second Equation
- Substitute y into the second equation:
2x sec θ - [(x sin θ) / (2 cos θ)] cosec θ = 3
- Simplifying the second term:
cosec θ = 1 / sin θ
Therefore,
y cosec θ = (x sin θ) / (2 cos θ) * (1 / sin θ) = (x) / (2 cos θ)
- The equation becomes:
2x sec θ - (x / (2 cos θ)) = 3
Step 3: Simplify the Equation
- Multiply through by 2 cos θ to eliminate the fraction:
4x cos θ sec θ - x = 6 cos θ
4x - x = 6 cos θ
3x = 6 cos θ
- Thus, x = 2 cos θ
Step 4: Find y in terms of x
- Substitute x back into the expression for y:
y = (2 cos θ sin θ) / (2 cos θ) = sin θ
Step 5: Find the Relation Between x and y
- Now we have:
x = 2 cos θ
y = sin θ
- Squaring both expressions gives:
x² = 4 cos² θ
y² = sin² θ
- Using the identity sin² θ + cos² θ = 1:
y² = 1 - cos² θ
- Therefore, we can express this as:
x² + 4y² = 4
Conclusion
- The final relation between x and y is:
x² + 4y² = 4 (Option B)
- We have two equations:
1. 2y cos θ = x sin θ
2. 2x sec θ - y cosec θ = 3
Step 1: Manipulate the First Equation
- From the first equation, we can express y in terms of x:
y = (x sin θ) / (2 cos θ)
Step 2: Substitute y in the Second Equation
- Substitute y into the second equation:
2x sec θ - [(x sin θ) / (2 cos θ)] cosec θ = 3
- Simplifying the second term:
cosec θ = 1 / sin θ
Therefore,
y cosec θ = (x sin θ) / (2 cos θ) * (1 / sin θ) = (x) / (2 cos θ)
- The equation becomes:
2x sec θ - (x / (2 cos θ)) = 3
Step 3: Simplify the Equation
- Multiply through by 2 cos θ to eliminate the fraction:
4x cos θ sec θ - x = 6 cos θ
4x - x = 6 cos θ
3x = 6 cos θ
- Thus, x = 2 cos θ
Step 4: Find y in terms of x
- Substitute x back into the expression for y:
y = (2 cos θ sin θ) / (2 cos θ) = sin θ
Step 5: Find the Relation Between x and y
- Now we have:
x = 2 cos θ
y = sin θ
- Squaring both expressions gives:
x² = 4 cos² θ
y² = sin² θ
- Using the identity sin² θ + cos² θ = 1:
y² = 1 - cos² θ
- Therefore, we can express this as:
x² + 4y² = 4
Conclusion
- The final relation between x and y is:
x² + 4y² = 4 (Option B)
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If 2y cos θ = x sin θ and 2x sec θ – y cosec θ = 3, then the relation between x and y is (SSC CGL 2nd Sit. 2012)a)2x2 + y2 = 2b)x2 + 4y2 = 4c)x2 + 4y2 = 1d)4x2 + y2 = 4Correct answer is option 'B'. Can you explain this answer?
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