X^2+9y^2-4x+54y+49=0 coordinates of the foci

Question

Graphing and finding the center and the foci.

Answer 1

The equation is .

The standard form of the equation of an ellipse center (h, k) with major and minor axes of lengths 2a and 2b (where 0 < b < a) is or .

The vertices and foci lie on the major axis, a and c units, respectively, from the center (h, k)  and the relation between a, b and c is c² = a² - b².

Write the given equation in standard form by completing the squares.

.

Because the denominator of the y² - term (4) is smaller than the denominator of the x² - term (36), the major axis is horizontal.

Compare the equation with .

a² = 36, b² = 4, h = 2 and k = - 3.

a = 6  and b = 2.

To find the value of c, substitute the value of a² = 36 and b² = 4 in c² = a² - b².

c² = 36 - 4 = 32

c = ± 4√2.

Center = (h, k ) = (2, - 3).

Foci = (h ± c, k ) = (2 ± 4√2, - 3).

Graph the ellipse :

The graph of the ellipse is translated 2 units to the right and down 3 units.

The center is at (2, - 3) and the foci are at (2 + 4√2, - 3) and (2 - 4√2, - 3).

From the graph, we can observe that, the length of the major axis is still 12 units, and the length of the minor axis is still 4 units.