The length of the subtangent to the ellipse `x=a cos t, y= b sin t ` at ` t = pi//4` is

Options

A. a
B. b
C. `b//sqrt2`
D. `a//sqrt2`

Answer

To find the length of the subtangent to the ellipse given by the parametric equations \( x = a \cos t \) and \( y = b \sin t \) at \( t = \frac{\pi}{4} \), we can follow these steps:

### Step 1: Find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)

Given:
- \( x = a \cos t \)
- \( y = b \sin t \)

We differentiate both with respect to \( t \):
\[ \frac{dx}{dt} = -a \sin t \]
\[ \frac{dy}{dt} = b \cos t \]

### Step 2: Find \( \frac{dy}{dx} \)

Using the chain rule, we can find \( \frac{dy}{dx} \) as follows:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{b \cos t}{-a \sin t} = -\frac{b}{a} \cdot \cot t \]

### Step 3: Evaluate \( \frac{dy}{dx} \) at \( t = \frac{\pi}{4} \)

Substituting \( t = \frac{\pi}{4} \):
\[ \cot\left(\frac{\pi}{4}\right) = 1 \]
Thus,
\[ \frac{dy}{dx} \bigg|_{t=\frac{\pi}{4}} = -\frac{b}{a} \cdot 1 = -\frac{b}{a} \]

### Step 4: Find the coordinates \( (x_1, y_1) \) at \( t = \frac{\pi}{4} \)

Now, we find the coordinates at \( t = \frac{\pi}{4} \):
\[ x_1 = a \cos\left(\frac{\pi}{4}\right) = a \cdot \frac{1}{\sqrt{2}} = \frac{a}{\sqrt{2}} \]
\[ y_1 = b \sin\left(\frac{\pi}{4}\right) = b \cdot \frac{1}{\sqrt{2}} = \frac{b}{\sqrt{2}} \]

### Step 5: Calculate the length of the subtangent

The formula for the length of the subtangent \( ST \) is given by:
\[ ST = \frac{y_1}{\frac{dy}{dx}} \quad \text{(taking the modulus)} \]
Substituting the values we have:
\[ ST = \frac{\frac{b}{\sqrt{2}}}{-\frac{b}{a}} = \frac{b}{\sqrt{2}} \cdot \left(-\frac{a}{b}\right) = -\frac{a}{\sqrt{2}} \]
Taking the modulus:
\[ ST = \frac{a}{\sqrt{2}} \]

### Final Answer:
The length of the subtangent to the ellipse at \( t = \frac{\pi}{4} \) is \( \frac{a}{\sqrt{2}} \).

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