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In a lemniscate curve, the angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve (i.e. straight) is
- a)equal to
- b)double
- c)three times
- d)four times
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
In a lemniscate curve, the angle between the tangent at the end of the...
Let P be a point on Laminscate with coordinates (μ, α).
When a tangent is drawn and it meets at P' to initial tangent, the angle between tangent and initial tangent is 'ϕ'
Where, β = The angle of polar ray op with tangent pp', and ∝ = polar deflection angle, ⇒ β = 2 α
The equation of laminscate is given by:
Differentiating w.r.t α. on both the sides,
We get,
∵ β = 2 α
We know ϕ = α + β = 2 α + α = 3 α
⇒ ϕ = 3 α
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In a lemniscate curve, the angle between the tangent at the end of the...
Introduction:
The lemniscate curve is a mathematical curve that resembles the shape of the number eight (∞). It is defined in polar coordinates by the equation r^2 = a^2 * cos(2θ), where r is the distance from the origin, a is a constant, and θ is the angle measured from the positive x-axis.
Understanding the problem:
In this problem, we are given a lemniscate curve and we need to find the angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve.
Approach:
To find the angle between two tangents, we need to determine the angles they make with the x-axis and then find the difference between these angles.
Determining the angles:
1. The tangent at the end of the polar ray:
- The polar ray is the line that starts from the origin and passes through the point where the lemniscate curve intersects the positive x-axis.
- Let's denote this point as P. The polar coordinate of P is (a, 0).
- To find the tangent at point P, we differentiate the equation of the lemniscate curve with respect to θ and evaluate it at θ = 0 (since P lies on the positive x-axis).
- Differentiating the equation r^2 = a^2 * cos(2θ) with respect to θ, we get 2r * dr/dθ = -2a^2 * sin(2θ).
- At θ = 0, r = a and sin(2θ) = sin(0) = 0. Therefore, the tangent at point P is vertical (parallel to the y-axis).
2. The tangent at the commencement of the curve:
- The commencement of the curve is the point where the lemniscate curve starts from the origin.
- Let's denote this point as Q. The polar coordinate of Q is (0, 0).
- To find the tangent at point Q, we need to determine the derivative of the lemniscate curve equation with respect to θ and evaluate it at θ = 0.
- Differentiating the equation r^2 = a^2 * cos(2θ) with respect to θ, we get 2r * dr/dθ = -2a^2 * sin(2θ).
- At θ = 0, r = 0 and sin(2θ) = sin(0) = 0. Therefore, the tangent at point Q is also vertical (parallel to the y-axis).
Calculating the angle:
Since both tangents are vertical, the angle between them is 0 degrees. However, in this problem, the options provided do not include 0 as a choice. Therefore, we need to consider the options and choose the closest one.
Conclusion:
The angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve is 0 degrees, which is not included in the given options.
The lemniscate curve is a mathematical curve that resembles the shape of the number eight (∞). It is defined in polar coordinates by the equation r^2 = a^2 * cos(2θ), where r is the distance from the origin, a is a constant, and θ is the angle measured from the positive x-axis.
Understanding the problem:
In this problem, we are given a lemniscate curve and we need to find the angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve.
Approach:
To find the angle between two tangents, we need to determine the angles they make with the x-axis and then find the difference between these angles.
Determining the angles:
1. The tangent at the end of the polar ray:
- The polar ray is the line that starts from the origin and passes through the point where the lemniscate curve intersects the positive x-axis.
- Let's denote this point as P. The polar coordinate of P is (a, 0).
- To find the tangent at point P, we differentiate the equation of the lemniscate curve with respect to θ and evaluate it at θ = 0 (since P lies on the positive x-axis).
- Differentiating the equation r^2 = a^2 * cos(2θ) with respect to θ, we get 2r * dr/dθ = -2a^2 * sin(2θ).
- At θ = 0, r = a and sin(2θ) = sin(0) = 0. Therefore, the tangent at point P is vertical (parallel to the y-axis).
2. The tangent at the commencement of the curve:
- The commencement of the curve is the point where the lemniscate curve starts from the origin.
- Let's denote this point as Q. The polar coordinate of Q is (0, 0).
- To find the tangent at point Q, we need to determine the derivative of the lemniscate curve equation with respect to θ and evaluate it at θ = 0.
- Differentiating the equation r^2 = a^2 * cos(2θ) with respect to θ, we get 2r * dr/dθ = -2a^2 * sin(2θ).
- At θ = 0, r = 0 and sin(2θ) = sin(0) = 0. Therefore, the tangent at point Q is also vertical (parallel to the y-axis).
Calculating the angle:
Since both tangents are vertical, the angle between them is 0 degrees. However, in this problem, the options provided do not include 0 as a choice. Therefore, we need to consider the options and choose the closest one.
Conclusion:
The angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve is 0 degrees, which is not included in the given options.
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In a lemniscate curve, the angle between the tangent at the end of the polar ray and the tangent at the commencement of the curve (i.e. straight) isa)equal tob)doublec)three timesd)four timesCorrect answer is option 'C'. Can you explain this answer?
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