Convert `2x^2+3y^2=6` into the polar equation.

Explanation

To convert the equation \(2x^2 + 3y^2 = 6\) into polar coordinates, we follow these steps:

### Step 1: Substitute Polar Coordinates
We know that in polar coordinates:
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)

Substituting these into the equation gives:
\[ 2(r \cos \theta)^2 + 3(r \sin \theta)^2 = 6 \]

### Step 2: Expand the Equation
Now, we expand the equation:
\[ 2(r^2 \cos^2 \theta) + 3(r^2 \sin^2 \theta) = 6 \]

### Step 3: Factor out \(r^2\)
We can factor out \(r^2\) from the left side:
\[ r^2(2 \cos^2 \theta + 3 \sin^2 \theta) = 6 \]

### Step 4: Isolate \(r^2\)
Next, we isolate \(r^2\) by dividing both sides by \((2 \cos^2 \theta + 3 \sin^2 \theta)\):
\[ r^2 = \frac{6}{2 \cos^2 \theta + 3 \sin^2 \theta} \]

### Step 5: Final Polar Equation
Thus, the polar equation is:
\[ r^2 = \frac{6}{2 \cos^2 \theta + 3 \sin^2 \theta} \]

### Summary
The final polar equation derived from \(2x^2 + 3y^2 = 6\) is:
\[ r^2 = \frac{6}{2 \cos^2 \theta + 3 \sin^2 \theta} \]

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