The `theta` eliminate of the following equations is `x = a cos ^(3) theta , y = b sin^(3) theta`

Options

A. `((a)/(x))^(2//3) + (( b)/( y ))^(2//3) =1`
B. `((b)/( x))^(2//3) + ((a)/( y ))^(2//3) = 1`
C. `((x)/( a))^(2//3) + (( y )/( b))^(2//3)=1`
D. `((x)/( b))^(2//3) +( (y)/( a))^(2//3) =1`

Answer

To eliminate the parameter \(\theta\) from the equations \(x = a \cos^3 \theta\) and \(y = b \sin^3 \theta\), we can follow these steps:

### Step 1: Rewrite the equations
We start with the given equations:
1. \(x = a \cos^3 \theta\) (Equation 1)
2. \(y = b \sin^3 \theta\) (Equation 2)

### Step 2: Express \(\cos^3 \theta\) and \(\sin^3 \theta\)
From Equation 1, we can express \(\cos^3 \theta\):
\[ \cos^3 \theta = \frac{x}{a} \]
From Equation 2, we can express \(\sin^3 \theta\):
\[ \sin^3 \theta = \frac{y}{b} \]

### Step 3: Take the cube root
To find \(\cos \theta\) and \(\sin \theta\), we take the cube root of both sides:
\[ \cos \theta = \left(\frac{x}{a}\right)^{\frac{1}{3}} \quad \text{(Equation 3)} \]
\[ \sin \theta = \left(\frac{y}{b}\right)^{\frac{1}{3}} \quad \text{(Equation 4)} \]

### Step 4: Square both equations
Now, we square both equations to eliminate the cube roots:
\[ \cos^2 \theta = \left(\frac{x}{a}\right)^{\frac{2}{3}} \quad \text{(Equation 5)} \]
\[ \sin^2 \theta = \left(\frac{y}{b}\right)^{\frac{2}{3}} \quad \text{(Equation 6)} \]

### Step 5: Add the two equations
Next, we add Equation 5 and Equation 6:
\[ \cos^2 \theta + \sin^2 \theta = \left(\frac{x}{a}\right)^{\frac{2}{3}} + \left(\frac{y}{b}\right)^{\frac{2}{3}} \]
Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), we have:
\[ 1 = \left(\frac{x}{a}\right)^{\frac{2}{3}} + \left(\frac{y}{b}\right)^{\frac{2}{3}} \]

### Step 6: Final equation
Rearranging gives us the final equation:
\[ \frac{x^{\frac{2}{3}}}{a^{\frac{2}{3}}} + \frac{y^{\frac{2}{3}}}{b^{\frac{2}{3}}} = 1 \]

This equation represents the relationship between \(x\) and \(y\) without the parameter \(\theta\).

### Summary
The final result is:
\[ \frac{x^{\frac{2}{3}}}{a^{\frac{2}{3}}} + \frac{y^{\frac{2}{3}}}{b^{\frac{2}{3}}} = 1 \]