The equation of tangent to the curve ` x=a cos^(3)t ,y=a sin^(3) t ` at 't' is

Options

A. `x sec t- y "cosec" t=a`
B. `x sec t+ y "cosec" t=a`
C. `x "cosec"t+y cos t=a`
D. `x sec t+ y cos t =-a `

Answer

To find the equation of the tangent to the curve defined by the parametric equations \( x = a \cos^3 t \) and \( y = a \sin^3 t \) at a specific parameter \( t \), we can follow these steps:

### Step 1: Differentiate \( x \) and \( y \) with respect to \( t \)

We start by differentiating \( x \) and \( y \) with respect to \( t \):

\[ \frac{dx}{dt} = \frac{d}{dt}(a \cos^3 t) = a \cdot 3 \cos^2 t \cdot (-\sin t) = -3a \cos^2 t \sin t \]

\[ \frac{dy}{dt} = \frac{d}{dt}(a \sin^3 t) = a \cdot 3 \sin^2 t \cdot \cos t = 3a \sin^2 t \cos t \]

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

Using the chain rule, we can find the slope of the tangent line:

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3a \sin^2 t \cos t}{-3a \cos^2 t \sin t} \]

This simplifies to:

\[ \frac{dy}{dx} = -\frac{\sin t}{\cos t} = -\tan t \]

### Step 3: Write the equation of the tangent line

The equation of the tangent line at the point \( (x_1, y_1) \) with slope \( m \) is given by:

\[ y - y_1 = m(x - x_1) \]

Here, \( x_1 = a \cos^3 t \), \( y_1 = a \sin^3 t \), and \( m = -\tan t \). Substituting these values, we have:

\[ y - a \sin^3 t = -\tan t (x - a \cos^3 t) \]

### Step 4: Rearranging the equation

Expanding this equation gives:

\[ y - a \sin^3 t = -\frac{\sin t}{\cos t} (x - a \cos^3 t) \]

Multiplying through by \( \cos t \):

\[ \cos t (y - a \sin^3 t) = -\sin t (x - a \cos^3 t) \]

Expanding both sides results in:

\[ y \cos t - a \sin^3 t \cos t = -\sin t x + a \sin t \cos^3 t \]

Rearranging gives:

\[ \sin t x + \cos t y = a (\sin t \cos^3 t + \sin^3 t \cos t) \]

### Step 5: Simplifying using trigonometric identities

Using the identity \( \sin^2 t + \cos^2 t = 1 \):

\[ \sin t \cos^3 t + \sin^3 t \cos t = \sin t \cos t (\cos^2 t + \sin^2 t) = \sin t \cos t \]

Thus, we can rewrite the equation as:

\[ \sin t x + \cos t y = a \sin t \cos t \]

### Final Equation

The final equation of the tangent line can be expressed as:

\[ \frac{x}{\cos t} + \frac{y}{\sin t} = a \]