If x = `a cos^(3) theta, y = a sin ^(3) theta` then `sqrt(1+((dy)/(dx))^(2))`=?

Options

A. `tan^(2) theta`
B. `sec ^(2) theta`
C. `sec theta`
D. `|sec theta|`

Answer

To solve the problem, we need to find the value of \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \) given the parametric equations \( x = a \cos^3 \theta \) and \( y = a \sin^3 \theta \).

### Step 1: Find \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \)

First, we differentiate \( x \) and \( y \) with respect to \( \theta \):

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

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

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

Using the chain rule, we can find \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{3a \sin^2 \theta \cos \theta}{-3a \cos^2 \theta \sin \theta} \]

### Step 3: Simplify \( \frac{dy}{dx} \)

Now we simplify the expression:

\[ \frac{dy}{dx} = \frac{3a \sin^2 \theta \cos \theta}{-3a \cos^2 \theta \sin \theta} = -\frac{\sin \theta}{\cos \theta} = -\tan \theta \]

### Step 4: Calculate \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \)

Now we substitute \( \frac{dy}{dx} \) into the expression we need to evaluate:

\[ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \sqrt{1 + (-\tan \theta)^2} = \sqrt{1 + \tan^2 \theta} \]

### Step 5: Use the identity \( 1 + \tan^2 \theta = \sec^2 \theta \)

Using the trigonometric identity:

\[ 1 + \tan^2 \theta = \sec^2 \theta \]

Thus, we have:

\[ \sqrt{1 + \tan^2 \theta} = \sqrt{\sec^2 \theta} = |\sec \theta| \]

### Final Answer

Therefore, the final answer is:

\[ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} = |\sec \theta| \]

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