If OT is the perpendicular drawn from the origin to the tangent at any point t to the curve `x = a cos^(3) t, y = a sin^(3) t`, then OT is equal to :

Options

A. `a sin 2t`
B. `(a)/(2) sin 2t`
C. `2a sin 2t`
D. 2a

Answer

To find the length of the perpendicular OT drawn from the origin to the tangent at any point T on the curve defined by the parametric equations \( x = a \cos^3 t \) and \( y = a \sin^3 t \), we can follow these steps:

### Step 1: Determine the coordinates of point T
The coordinates of point T on the curve are given by:
\[ T(a \cos^3 t, a \sin^3 t) \]

### Step 2: Find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)
Using the chain rule, we differentiate \( x \) and \( y \):
\[ \frac{dx}{dt} = \frac{d}{dt}(a \cos^3 t) = -3a \cos^2 t \sin t \]
\[ \frac{dy}{dt} = \frac{d}{dt}(a \sin^3 t) = 3a \sin^2 t \cos t \]

### Step 3: Find the slope of the tangent line
The slope \( m \) of the tangent line at point T is given by:
\[ m = \frac{dy/dt}{dx/dt} = \frac{3a \sin^2 t \cos t}{-3a \cos^2 t \sin t} = -\frac{\sin t}{\cos t} = -\tan t \]

### Step 4: Write the equation of the tangent line
Using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), we substitute \( (x_1, y_1) = (a \cos^3 t, a \sin^3 t) \) and \( m = -\tan t \):
\[ y - a \sin^3 t = -\tan t (x - a \cos^3 t) \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ y = -\tan t \cdot x + a \sin^3 t + a \cos^3 t \tan t \]
This can be rewritten as:
\[ \tan t \cdot x + y = a(\sin^3 t + \cos^3 t) \]

### Step 6: Identify coefficients for the perpendicular distance formula
The equation can be expressed in the form \( Ax + By + C = 0 \):
\[ \tan t \cdot x + y - a(\sin^3 t + \cos^3 t) = 0 \]
Here, \( A = \tan t \), \( B = 1 \), and \( C = -a(\sin^3 t + \cos^3 t) \).

### Step 7: Calculate the perpendicular distance from the origin to the line
The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_0, y_0) = (0, 0) \):
\[ OT = \frac{|0 + 0 - a(\sin^3 t + \cos^3 t)|}{\sqrt{(\tan t)^2 + 1}} = \frac{a(\sin^3 t + \cos^3 t)}{\sqrt{\tan^2 t + 1}} = \frac{a(\sin^3 t + \cos^3 t)}{\sec t} \]

### Step 8: Simplify the expression
Since \( \sec t = \frac{1}{\cos t} \):
\[ OT = a(\sin^3 t + \cos^3 t) \cos t \]

### Step 9: Use the identity for \( \sin^3 t + \cos^3 t \)
Using the identity \( \sin^3 t + \cos^3 t = (\sin t + \cos t)(\sin^2 t - \sin t \cos t + \cos^2 t) \):
\[ OT = a(\sin t + \cos t)(1 - \sin t \cos t) \cos t \]

### Step 10: Final expression for OT
Using the double angle identity \( \sin 2t = 2 \sin t \cos t \):
\[ OT = \frac{a}{2} \sin 2t \]

### Final Result
Thus, the length of OT is:
\[ OT = \frac{a}{2} \sin 2t \]