Question

Write the equation below in standard form and then answer the following questions. If a value is a non-integer type your answer as a decimal rounded to the hundredths place. 16x^2+64x+4y^2-8y+4=0The center is (h,k). h= Answer and k= AnswerThe value for a is Answer . The value for b is Answer .The foci with the positive y value is the point ( Answer, Answer)The foci with the negative y value is the point ( Answer, Answer)

Answer 1

The characteristics of the ellipse are as follows:

- The center is
`\( (h, k) \)` where
`\( h = -2 \)` and
`\( k = 1 \)`.

- The value for
`\( a \)` (the semi-minor axis) is approximately
`\( 1.12 \)`.

- The value for
`\( b \)` (the semi-major axis) is approximately
`\( 2.24 \)`.

- The foci with the positive
`\( y \)` value is the point
`\( (-2, 2.94) \)`, rounded to two decimal places.

- The foci with the negative
`\( y \)` value is the point
`\( (-2, -0.94) \)`, rounded to two decimal places.

To write the given equation
`\(16x^2 + 64x + 4y^2 - 8y + 4 = 0\)` in standard form and determine the various characteristics of the conic section it represents, we will complete the square for both \(x\) and \(y\) terms. Here are the steps:

1. Group the
`\(x\)` and
`\(y\)` terms together:

`\[ 16x^2 + 64x + 4y^2 - 8y = -4 \]`

2. Factor out the coefficients of the squared terms from each group:

`\[ 16(x^2 + 4x) + 4(y^2 - 2y) = -4 \]`

3. Complete the square for the \(x\) terms and the \(y\) terms. This involves adding the square of half the coefficient of the \(x\) term inside the parentheses to both sides of the equation, and similarly for the \(y\) term:

`\[ 16(x^2 + 4x + (4/2)^2) + 4(y^2 - 2y + (2/2)^2) = -4 + 16(4/2)^2 + 4(2/2)^2 \]`

4. Simplify and convert the equation into standard form:

`\[ 16(x + 2)^2 + 4(y - 1)^2 = 16 + 4 \]`

5. Continue simplifying:

`\[ 16(x + 2)^2 + 4(y - 1)^2 = 20 \]`

6. Divide through by 20 to get the equation in standard form:

`\[ ((x + 2)^2)/(20/16) + ((y - 1)^2)/(20/4) = 1 \]`

`\[ ((x + 2)^2)/(5/4) + ((y - 1)^2)/(5) = 1 \]`

The standard form of an ellipse is
`\( ((x - h)^2)/(a^2) + ((y - k)^2)/(b^2) = 1 \)`, which allows us to identify the center \((h, k)\), and the values of \(a\) and \(b\).

So, for our equation:

-
`\( h = -2 \)`

-
`\( k = 1 \)`

-
`\( a^2 = 5/4 \)` which gives us
`\( a = √(5/4) \)`

-
`\( b^2 = 5 \)` which gives us
`\( b = √(5) \)`

For the foci of an ellipse, which are located along the major axis, we use the relationship
`\( c^2 = a^2 - b^2 \)` (or vice versa depending on which is larger). For this ellipse, since
`\( a^2 < b^2 \)`, we'll have
`\( c^2 = b^2 - a^2 \)`.

Let's calculate the exact values for
`\( a \)`,
`\( b \)`, and
`\( c \)`, and then determine the coordinates of the foci.

The characteristics of the ellipse are as follows:

- The center is
`\( (h, k) \)` where
`\( h = -2 \)` and
`\( k = 1 \)`.

- The value for
`\( a \)` (the semi-minor axis) is approximately
`\( 1.12 \)`.

- The value for
`\( b \)` (the semi-major axis) is approximately
`\( 2.24 \)`.

- The foci with the positive
`\( y \)` value is the point
`\( (-2, 2.94) \)`, rounded to two decimal places.

- The foci with the negative
`\( y \)` value is the point
`\( (-2, -0.94) \)`, rounded to two decimal places.

Answer 2

Step-by-step explanation:

Given the function:

`16x^2+64x+4y^2-8y+4=0`

This can be written as:

`16(x+2)^2+4(y-1)^2=8^2`

The center here is:

(h, k) = (-2, 1)

The radius is