Precalculus Help??? Graph the ellipse???

Question

Graph the ellipse.

x² + 16y² − 224y + 768 = 0

Specify the lengths of the major and minor axes, the foci, and the eccentricity.

length of major axis
length of minor axis
foci	(x, y) =		(smaller x-value)
	(x, y) =		(larger x-value)
eccentricity

Specify the center of the ellipse.
(x, y) = 

 

Answer 1

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 (x - h)²/a² + (y - k)²/b² = 1(major axis is horizontal) or (x - h)²/b² + (y - k)²/a² = 1(major axis is vertical).

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.

The conic equation is x² + 16y² − 224y + 768 = 0.

Complete the square to write the equation in standard form.

x² + 16(y² − 14y) + 768 = 0

To change the expressions y² − 14y  into a perfect square trinomial add (half the y coefficient)² to each side of the expression

 Here y coefficient = - 14. so, (half the y coefficient)² = (- 14/2)² = = (- 7)² = 49.

Add 16(49) to each side.

x² + 16(y² − 14y + 49) + 768 = 16(49)

x² + 16(y − 7)² + 768 = 784

x² + 16(y − 7)² = 16

x²/16 + [16(y − 7)²]/16 = 16/16

(x - 0)²/4² + (y - 7)²/1² = 1.

Compare the above equation with (x - h)²/a² + (y - k)²/b² = 1(major axis is horizontal).

h = 0, k = 7, a = 4 and b = 1.

Center = (h, k) = (0, 7)

Length of Major Axis = 2a = 2(4) = 8 units.

Length of Minor Axis = 2b = 2(1) = 2 units.

c² = a² - b² = (4)² - (1)² = 16 - 1 = 15.

c = ± √(15)

Foci = (h ± c, k) = (0 ± √(15), 7).

The foci are (- √(15), 7) and (√(15), 7).

The eccentricity e = c/a = √(15)/4.

From  this  standard  form, it  follows  that  the  center  is (h, k) = (0, 7). Because  the denominator of the x-term is a² = 4²,  the endpoints of the major axis lie four units to the  right  and  left  of  the  center. Then the points are (-4, 7) and (4, 7).  Similarly, because  the  denominator  of  the y-term  is b² = 1² the endpoints of the minor axis lie one unit up and down from the center. Then the points are (0, 8) and (0, 6).The ellipse is shown in Figure.