X^2 + 2Y^2 + 2X - 20Y + 43=0 Find the foci and vertices

Question

My friends and I keep getting squareroot of 8 as c but wee know that isnt right. pleace help us

Answer 1

Hello Henry,

 

Complete the square to get the equation into graphing form:

 

x² + 2y² + 2x - 20y + 43 = 0

(x² + 2x +1) + 2(y² - 10y + 25) = -43 + 1 + 50

(x + 1)² + 2(y - 5)² = 8

 

(x + 1)² / 8 + (y - 5)² / 4 = 1

 

In an ellipse, a² is always the larger of the two denominators and it is always under the major axis.

 

In this case, a² is 8 and x is the major axis.

 

a² = 8

a = 2√2

 

b² = 4

b = 2

 

c² = a² - b²

c² = 8 - 4

c² = 4

c = 2

 

Center (-1, 5)

 

To find the vertices: add and subtract "a" to the "major axis side" of the center. Since x is the major axis, we need to add and subtract 2√2 to the x coordinate of the center.

 

Vertices (-1 + 2√2, 5), (-1 - 2√2, 5)

 

To find the foci: add and subtract "c" to the "major axis side" of the center. Since x is the major axis, we need to add and subtract 2 to the x coordinate of the center.

 

Foci (1,5), (-3,5)

 

I hope this helps!

Answer 2

Complete the two squares

(x² + 2x) + 2(y² -10y) = -43

(x²+ 2x + 1) + 2(y² - 10y + 25) = -43 + 1 + 50

(x + 1)² + 2(y - 5)² = 8

 

(x + 1)²/8 + (y - 5)²/4 = 1

 

 

Center (-1,5) 

orientation: horizontal

 

a = 2√2

b = 2

 

c = √[(2√2)² - (2)²] = √(8 - 4) = √4 = 2

 

 

foci (-3,5) (1,5)