Conic Sections?

Question

Please show work with explanations. Thank you!!

Answer 1

The conic equation is

The coefficients of x ² and y ² are of the same sign but unequal coefficients ,the graph will be an ellipse.

By using completing squre method convert the equation in standardform.

To change the expressions (x ² + 8x ) and (y ² + 8y ) into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)²to each side of the equation.

The standard form for an ellipse is in a form = 1, So divide both sides of equation by 225 to set it equal to 1.

Compare it to standard form of ellipse

a ² > b ²

If the larger denominator is under the "x" term, then the ellipse is horizontal.

center (h, k ) = (- 4, - 4)

a  = length of semi-major axis = 5

b  = length of semi-minor axis = 3.

c is the distance from the center to each focus.

Foci: (h + c , k ), (h - c, k )

(- 4 + 4, - 4), (- 4 - 4, -4) = (0, - 4)(- 8, - 4)

Vertices: (h + a, k) (h - a, k)

(-4 + 5, - 4 ) (- 4 - 5, - 4)

= (1, - 4) (- 9, - 4)

Therefore, C : (-4,-4), F : (0,-4), (-8, -4) and V : (1, -4), ( - 9, - 4)

Option D is correct choice.