Find the Foci x^2+16y^2=16

Solution:

Step 1: Convert to the standard form of the ellipse.

Step 1.1: Normalize the equation by dividing all terms by 16.

$$\frac{x^2}{16} + \frac{16y^2}{16} = \frac{16}{16}$$

Step 1.2: Simplify the equation to set the right side to 1, which is required for the standard form.

$$\frac{x^2}{16} + y^2 = 1$$

Step 2: Identify the ellipse's form to determine the center, major axis, and minor axis.

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

Step 3: Align the values with the standard form to find the major axis (a), minor axis (b), and the offsets from the origin (h and k).

$$a = 4, b = 1, h = 0, k = 0$$

Step 4: Calculate c, the distance from the center to a focus.

Step 4.1: Use the formula to find the distance from the center to a focus.

$$c = \sqrt{a^2 - b^2}$$

Step 4.2: Plug in the values for a and b.

$$c = \sqrt{4^2 - 1^2}$$

Step 4.3: Perform the simplification.

Step 4.3.1: Square the value of 4.

$$c = \sqrt{16 - 1^2}$$

Step 4.3.2: Recognize that any number raised to the power of 1 is itself.

$$c = \sqrt{16 - 1}$$

Step 4.3.3: Multiply -1 by 1.

$$c = \sqrt{15}$$

Step 4.3.4: Subtract 1 from 16.

$$c = \sqrt{15}$$

Step 5: Determine the foci of the ellipse.

Step 5.1: The first focus is found by adding c to h.

$$(h + c, k)$$

Step 5.2: Insert the known values for h, c, and k.

$$(0 + \sqrt{15}, 0)$$

Step 5.3: Simplify the expression.

$$(\sqrt{15}, 0)$$

Step 5.4: The second focus is found by subtracting c from h.

$$(h - c, k)$$

Step 5.5: Insert the known values for h, c, and k.

$$(0 - \sqrt{15}, 0)$$

Step 5.6: Simplify the expression.

$$(-\sqrt{15}, 0)$$

Step 5.7: Note that ellipses have two foci.

$$(\text{Focus})_1: (\sqrt{15}, 0)$$ $$(\text{Focus})_2: (-\sqrt{15}, 0)$$

Knowledge Notes:

To solve for the foci of an ellipse given by the equation $x^2 + 16y^2 = 16$, we follow these steps:

1. Convert the equation to the standard form of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ by dividing all terms by the same number to get the right side equal to 1.

2. Identify the values of $a^2$, $b^2$, $h$, and $k$ from the standard form, where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $(h, k)$ is the center of the ellipse.

3. Calculate the distance $c$ from the center to the foci using the formula $c = \sqrt{a^2 - b^2}$.

4. The foci of the ellipse are located at $(h + c, k)$ and $(h - c, k)$.

Relevant knowledge points include:

• The standard form of an ellipse is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$.

• The semi-major axis $a$ is the longest radius of the ellipse, and the semi-minor axis $b$ is the shortest radius.

• The distance $c$ is related to $a$ and $b$ by the equation $c = \sqrt{a^2 - b^2}$.

• The foci of an ellipse are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant.