Exercise 37 Page 648

First, we need to rewrite this equation such that it fits the general equation of an ellipse.

Horizontal Ellipse : Vertical Ellipse : \frac{( x - h ) ^{2}}{a ^{2}} + \frac{( y - k ) ^{2}}{b ^{2}} = 1 \frac{( y - k ) ^{2}}{a ^{2}} + \frac{( x - h ) ^{2}}{b ^{2}} = 1

Because the denominator of the

y -

variable is greater than the denominator of the

x -

variable, the equation represents a vertical ellipse. Let's rewrite the equation a bit to make it easier to identify the values of

h,

k,

a,

and

b .

\frac{x ^{2}}{9} + \frac{y ^{2}}{3 6} = 1

▼

Simplify left-hand side

Write as a power

\frac{x ^{2}}{3 ^{2}} + \frac{y ^{2}}{6 ^{2}} = 1

Write as a difference

\frac{( x - 0 ) ^{2}}{3 ^{2}} + \frac{( y - 0 ) ^{2}}{6 ^{2}} = 1

Commutative Property of Addition

\frac{( y - 0 ) ^{2}}{6 ^{2}} + \frac{( x - 0 ) ^{2}}{3 ^{2}} = 1

Let's now use the equation to find the desired information.

Equation of the Vertical Ellipse	$\frac{( y - k ) ^{2}}{a ^{2}} + \frac{( x - h ) ^{2}}{b ^{2}} = 1,$ $a$ and $b$ positive, with $a > b$	$\frac{( y - 0 ) ^{2}}{6 ^{2}} + \frac{( x - 0 ) ^{2}}{3 ^{2}} = 1$
Center	$(h, k)$	$(0, 0)$
Length of Major Axis	$2 a$ units	$2 (6) = 12$ units
Length of Minor Axis	$2 b$ units	$2 (3) = 6$ units
Foci	$(h, k \pm c),$ $c^{2} = a^{2} - b^{2}$	$(0, 0 \pm c),$ $c^{2} = 6^{2} - 3^{2}$