Write an equation for an ellipse.

Question

Write an equation for the given ellipse that satisfies the following conditions. Center at (4,1); minor axis vertical, with length 8; c=3.

Answer 1

Since the minor axis is vertical, the equation has the form:

 

(x-4)²/a² + (y-1)²/b² = 1, where c² = a² - b²

 

Length of minor axis = 2b = 8.  So, b = 4

 

Since c = 3 and b = 4, 3² = a² - 4²

 

                                a² = 25   a = 5

 

Equation:  (x-4)²/25 + (y-1)²/16 = 1

 

 

Answer 2

The generic equation for an ellipse is:

 

(x/a)² + (y/b)² = 1, where a is the distance from the center of the ellipse to the horizontal vertex and b is the distance from the center of the ellipse to the vertical vertex.

 

the minor vertex is vertical with length 8. Therefore 2b = 8 -> b = 4. 

 

The linear eccentricity of the ellipse is c = 3. The relationship of the eccentricity to the major and minor axes is given by the equation:

 

c² = a² - b²

 

We need to find the value for the major axis a, as the eccentricity and major axis are given.

 

solving for a in the above equation -> a = √(c² + b²) = √(3^2 + 4^2) = 5.

 

Thus, the equation for the ellipse centered at the origin is:

 

(x/5)² + (y/4)² = 1

 

However, the center of the ellipse is actually centered at (4,1)

so to move the ellipse we simply translate the graph along the x and y 
directions by subtracting distance from x and y.

 

((x - 4)/5)² + ((y - 1)/4)² = 1.

 

Final Answer:

 

(x - 4)²/25 + (y - 1)²/16 = 1.