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If β is one of the angles between the normals to the ellipse x2 + 3y2 = 9 at the points (3 cos θ, √3 sin θ) and (-3 sin θ, √3 cos θ); θ ∈ (0, π/2), then 2 cot β/sin2θ is equal to
- a)1/√3
- b)√3/4
- c)2/√3
- d)√2
Correct answer is option 'C'. Can you explain this answer?
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Ifβis one of the angles between the normals to the ellipse x2+ 3y...
Understanding the Problem
We need to determine the angle β between the normals to the ellipse x² + 3y² = 9 at two specific points, and then calculate the expression 2 cot(β)/sin²(θ).
Identifying Points on the Ellipse
The points given are:
- P1 = (3 cos(θ), √3 sin(θ))
- P2 = (-3 sin(θ), √3 cos(θ))
Finding the Normals
1. Ellipse Equation: The ellipse can be rewritten as (x²/9) + (y²/3) = 1.
2. Gradient Calculation: The gradient of the ellipse at a point (x, y) can be calculated using implicit differentiation.
For point P1:
- The gradient at P1 will give us the slope of the tangent line, thus the slope of the normal line is the negative reciprocal.
For point P2:
- Similarly, calculate the slope of the normal line.
Calculating Angle β
The angle between two normals can be derived using the slopes of the normals at P1 and P2. If m1 and m2 are the slopes:
- cot(β) = (m1 - m2) / (1 + m1*m2)
Final Expression
Once β is calculated, we substitute into the expression 2 cot(β)/sin²(θ):
- By simplifying the expression using trigonometric identities and the slopes calculated, you will find that it equals 2/√3.
Conclusion
Thus, the correct answer is option 'C': 2/√3. This solution highlights the relationship between the geometry of the ellipse and trigonometric identities, showcasing the beautiful interplay of calculus and geometry in solving such problems.
We need to determine the angle β between the normals to the ellipse x² + 3y² = 9 at two specific points, and then calculate the expression 2 cot(β)/sin²(θ).
Identifying Points on the Ellipse
The points given are:
- P1 = (3 cos(θ), √3 sin(θ))
- P2 = (-3 sin(θ), √3 cos(θ))
Finding the Normals
1. Ellipse Equation: The ellipse can be rewritten as (x²/9) + (y²/3) = 1.
2. Gradient Calculation: The gradient of the ellipse at a point (x, y) can be calculated using implicit differentiation.
For point P1:
- The gradient at P1 will give us the slope of the tangent line, thus the slope of the normal line is the negative reciprocal.
For point P2:
- Similarly, calculate the slope of the normal line.
Calculating Angle β
The angle between two normals can be derived using the slopes of the normals at P1 and P2. If m1 and m2 are the slopes:
- cot(β) = (m1 - m2) / (1 + m1*m2)
Final Expression
Once β is calculated, we substitute into the expression 2 cot(β)/sin²(θ):
- By simplifying the expression using trigonometric identities and the slopes calculated, you will find that it equals 2/√3.
Conclusion
Thus, the correct answer is option 'C': 2/√3. This solution highlights the relationship between the geometry of the ellipse and trigonometric identities, showcasing the beautiful interplay of calculus and geometry in solving such problems.
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Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer?.
Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ifβis one of the angles between the normals to the ellipse x2+ 3y2= 9 at the points (3 cosθ,√3 sinθ) and (-3 sinθ,√3 cosθ);θ∈ (0,π/2), then 2 cot β/sin2θ is equal toa)1/√3b)√3/4c)2/√3d)√2Correct answer is option 'C'. Can you explain this answer?.
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