Find the standard form of the equation of the ellipse with the given characteristics.
Center: $( 0,4 )$ ;a = 2c;Vertices: $( - 8,8 ) , ( 8,8 )$ ,where a - thr semimajor axis,b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .

A) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { x ^ { 2 } } { 64 } - \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
C) $\frac { x ^ { 2 } } { 64 } - \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics. ​
Center: $( 0,4 )$ ;a = 2c;Vertices: $( - 8,8 ) , ( 8,8 )$ ,where a - thr semimajor axis,b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .
​
A) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { x ^ { 2 } } { 64 } - \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
C) $\frac { x ^ { 2 } } { 64 } - \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$

Quizplus · Accepted Answer

We can determine the center of the ellipse as the midpoint of the two vertices, which gives us:

Center: (11ea99db_b844_69ac_94ac_31ac3e1bca7a_TB4590_11 + 11ea99db_b844_69ad_94ac_9742b6fe1181_TB4590_11)/2 = 11ea99db_b844_69ac_94ac_c2ee734f619b_TB4590_11

We know that a = 2c, so we can use the distance formula to find the value of c (the distance from the center to a focus):

2c = distance between vertices/2 = sqrt((11ea99db_b844_69ad_94ac_9742b6fe1181_TB4590_11 - 11ea99db_b844_69ac_94ac_31ac3e1bca7a_TB4590_11)^2)/2 = 2sqrt(5)

c = sqrt(5)

We also know that a = distance between vertices/2 = sqrt((11ea99db_b844_69ad_94ac_9742b6fe1181_TB4590_11 - 11ea99db_b844_69ac_94ac_31ac3e1bca7a_TB4590_11)^2)/2 = 2

So, we can now use the standard form of the equation for an ellipse (with center at (h,k)):

((x - h)^2)/(a^2) + ((y - k)^2)/(b^2) = 1

Plugging in our values, we get:

((x - 11ea99db_b844_69ac_94ac_c2ee734f619b_TB4590_11)^2)/(2^2) + ((y - 11ea99db_b844_90be_94ac_ad8777db9d29_TB4590_11)^2)/(sqrt(5)^2) = 1

Simplifying and converting to the given format, we get:

Answer: D) 11ea99db_b844_b7d2_94ac_9995f34ea3d3_TB4590_11

Find the Standard Form of the Equation of the Ellipse (0,4)( 0,4 )(0,4)

Find the Standard Form of the Equation of the Ellipse $( 0,4 )$