Given 0 ≤ θ < 2π, find the range of values of θ for which two parabolas y = 2sqrt(3)(x - cosθ)^2 + sinθ and y = -2sqrt(3)(x + cosθ)^2 - sinθ intersect at two different points.

We are tasked with finding the range of values of \( \theta \) for which the two parabolas:

\[
y_1 = 2\sqrt{3}(x - \cos\theta)^2 + \sin\theta
\]
\[
y_2 = -2\sqrt{3}(x + \cos\theta)^2 - \sin\theta
\]

intersect at two different points, given that \( 0 \leq \theta < 2\pi \).

### Step 1: Understand the Geometry of the Parabolas

- The first parabola, \( y_1 \), is centered at \( x = \cos\theta \) and opens upwards.
- The second parabola, \( y_2 \), is centered at \( x = -\cos\theta \) and opens downwards.

Thus, the parabolas are symmetric about the origin (since their centers are at \( x = \pm \cos\theta \)) and open in opposite directions.

### Step 2: Set the Parabolas Equal to Each Other

To find the points of intersection, we set \( y_1 = y_2 \):

\[
2\sqrt{3}(x - \cos\theta)^2 + \sin\theta = -2\sqrt{3}(x + \cos\theta)^2 - \sin\theta
\]

### Step 3: Simplify the Equation

First, move all terms involving \( x \) and \( \sin\theta \) to one side:

\[
2\sqrt{3}(x - \cos\theta)^2 + 2\sqrt{3}(x + \cos\theta)^2 = -2\sin\theta
\]

Factor out \( 2\sqrt{3} \) on the left-hand side:

\[
2\sqrt{3} \left( (x - \cos\theta)^2 + (x + \cos\theta)^2 \right) = -2\sin\theta
\]

Expand the squares:

\[
(x - \cos\theta)^2 = x^2 - 2x\cos\theta + \cos^2\theta
\]
\[
(x + \cos\theta)^2 = x^2 + 2x\cos\theta + \cos^2\theta
\]

Now substitute these into the equation:

\[
2\sqrt{3} \left( (x^2 - 2x\cos\theta + \cos^2\theta) + (x^2 + 2x\cos\theta + \cos^2\theta) \right) = -2\sin\theta
\]

Simplify the terms inside the parentheses:

\[
(x^2 - 2x\cos\theta + \cos^2\theta) + (x^2 + 2x\cos\theta + \cos^2\theta) = 2x^2 + 2\cos^2\theta
\]

Thus, the equation becomes:

\[
2\sqrt{3} (2x

Explore how to find the range of θ where the parabolas y = 2√3(x - cosθ)^2 + sinθ and y = -2√3(x + cosθ)^2 - sinθ intersect at two points. This problem involves algebra, geometry, and trigonometry concepts and is suitable for undergraduate students.

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