The maximum distance from origin of a point on the curve `x=asint-bsin((a t)/b), y=acost-bcos((a t)/b)`, borth a,b>0 is

Options

A. a-b
B. a+b
C. `sqrt(a^2+b^2)`
D. `sqrt(a^2-b^2)`

Answer

To find the maximum distance from the origin of a point on the curve defined by the equations \( x = a \sin t - b \sin\left(\frac{a t}{b}\right) \) and \( y = a \cos t - b \cos\left(\frac{a t}{b}\right) \), we will follow these steps:

### Step 1: Write the distance formula
The distance \( d \) from the origin to a point \( (x, y) \) is given by:
\[ d = \sqrt{x^2 + y^2} \]
To simplify our calculations, we will maximize \( d^2 \) instead:
\[ d^2 = x^2 + y^2 \]

### Step 2: Substitute the expressions for \( x \) and \( y \)
Substituting the given expressions for \( x \) and \( y \):
\[ d^2 = \left(a \sin t - b \sin\left(\frac{a t}{b}\right)\right)^2 + \left(a \cos t - b \cos\left(\frac{a t}{b}\right)\right)^2 \]

### Step 3: Expand \( d^2 \)
Expanding both squares:
\[ d^2 = \left(a^2 \sin^2 t - 2ab \sin t \sin\left(\frac{a t}{b}\right) + b^2 \sin^2\left(\frac{a t}{b}\right)\right) + \left(a^2 \cos^2 t - 2ab \cos t \cos\left(\frac{a t}{b}\right) + b^2 \cos^2\left(\frac{a t}{b}\right)\right) \]
Combining the terms:
\[ d^2 = a^2 (\sin^2 t + \cos^2 t) + b^2 \left(\sin^2\left(\frac{a t}{b}\right) + \cos^2\left(\frac{a t}{b}\right)\right) - 2ab \left(\sin t \sin\left(\frac{a t}{b}\right) + \cos t \cos\left(\frac{a t}{b}\right)\right) \]

### Step 4: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ d^2 = a^2 + b^2 - 2ab \left(\sin t \sin\left(\frac{a t}{b}\right) + \cos t \cos\left(\frac{a t}{b}\right)\right) \]
Using the cosine of the difference identity:
\[ d^2 = a^2 + b^2 - 2ab \cos\left(t - \frac{a t}{b}\right) \]

### Step 5: Maximize \( d^2 \)
To maximize \( d^2 \), we need to minimize \( \cos\left(t - \frac{a t}{b}\right) \). The minimum value of \( \cos \theta \) is \(-1\). Thus:
\[ d^2_{\text{max}} = a^2 + b^2 + 2ab = (a + b)^2 \]

### Step 6: Calculate the maximum distance \( d \)
Taking the square root to find the maximum distance:
\[ d_{\text{max}} = \sqrt{(a + b)^2} = a + b \]

### Final Answer
The maximum distance from the origin of a point on the curve is:
\[ \boxed{a + b} \]