What are the foci of the ellipse represented by the equation 25x² + 16y² = 400?

To find the foci of the ellipse given by the equation 25x² + 16y² = 400, we first need to rewrite it in the standard form of an ellipse equation.

The general form of an ellipse centered at the origin is:

• Ellipse equation: (x²/a²) + (y²/b²) = 1

From the given equation, we can simplify it:

1. Divide every term by 400:
2. That gives us: 25x²/400 + 16y²/400 = 1
3. Which simplifies to: (x²/16) + (y²/25) = 1

Now, we can see that:

• a² = 25 => a = 5
• b² = 16 => b = 4

Next, since this is a vertical ellipse (because b² > a²), the foci can be calculated using the formula:

• c = √(b² – a²)

Substituting the values:

• c = √(25 – 16) = √9 = 3

Now, the coordinates of the foci (F₁ and F₂) for a vertical ellipse are:

• (0, c) => (0, 3)
• (0, -c) => (0, -3)

Therefore, the foci of the ellipse represented by the equation 25x² + 16y² = 400 are located at:

• (0, 3)
• (0, -3)