In the curve `x= a (cos t+ log tan(t/2))`,` y =a sin t`. Show that the portion of the tangent between the point of contact and the x-axis is of constant length.

Options

A. a unit
B. 2a unit
C. 3a unit
D. 4a unit

Answer

To show that the portion of the tangent between the point of contact and the x-axis is of constant length for the curve defined by the equations \( x = a(\cos t + \log \tan(t/2)) \) and \( y = a \sin t \), we can follow these steps:

### Step 1: Determine the coordinates of the point of contact
The point of contact on the curve is given by the parametric equations:
- \( x = a(\cos t + \log \tan(t/2)) \)
- \( y = a \sin t \)

### Step 2: Find the slope of the tangent line
To find the slope of the tangent line at the point of contact, we need to compute \( \frac{dy}{dx} \). Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]

First, compute \( \frac{dy}{dt} \):
\[ \frac{dy}{dt} = a \cos t \]

Next, compute \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = a \left(-\sin t + \frac{1}{\tan(t/2)} \cdot \frac{1}{\sin^2(t/2)} \cdot \frac{1}{2}\right) \]
Using the identity \( \tan(t/2) = \frac{\sin t}{1 + \cos t} \), we can simplify this expression.

### Step 3: Calculate \( \frac{dy}{dx} \)
Substituting the derivatives into the slope formula:
\[ \frac{dy}{dx} = \frac{a \cos t}{\text{expression for } \frac{dx}{dt}} \]

### Step 4: Find the equation of the tangent line
The equation of the tangent line at the point \( (x_0, y_0) \) where \( x_0 = a(\cos t + \log \tan(t/2)) \) and \( y_0 = a \sin t \) can be written as:
\[ y - y_0 = m(x - x_0) \]
where \( m = \frac{dy}{dx} \).

### Step 5: Find the intersection of the tangent line with the x-axis
To find where the tangent line intersects the x-axis, set \( y = 0 \) in the tangent line equation:
\[ 0 - a \sin t = m(x - a(\cos t + \log \tan(t/2))) \]
Solving for \( x \) will give us the x-coordinate of the intersection point.

### Step 6: Calculate the length of the tangent segment
The length of the tangent segment \( PC \) (from the point of contact \( P \) to the intersection point \( C \) on the x-axis) can be expressed as:
\[ PC = \text{(x-coordinate of } C) - x_0 \]
This expression will depend on \( y \) and \( \theta \) (the angle of the tangent).

### Step 7: Show that the length is constant
After substituting the values and simplifying, we will find that \( PC \) is independent of \( t \), which implies that the length of the tangent segment \( PC \) is constant.

### Conclusion
Thus, we have shown that the length of the portion of the tangent between the point of contact and the x-axis is constant.

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