Find the Properties (x^2)/4+(y^2)/9=1

Problem

(x2)/4+(y2)/9=1

Solution

1. Identify the type of conic section by observing the equation is in the form (x2)/(b2)+(y2)/(a2)=1 where a>b which represents a vertical ellipse.

2. Determine the center by noting that x and y are not shifted, meaning the center (h,k) is at the origin.

(h,k)=(0,0)

3. Calculate the semi-major and semi-minor axes by taking the square roots of the denominators.

a2=9⇒a=3

b2=4⇒b=2

4. Find the vertices by moving a units up and down from the center along the y-axis.

Vertices=(0,±3)

5. Find the co-vertices by moving b units left and right from the center along the x-axis.

Co-vertices=(±2,0)

6. Calculate the focal distance c using the relationship c2=a2−b2

c2=9−4=5

c=√(,5)

7. Identify the foci by moving c units up and down from the center along the major axis.

Foci=(0,±√(,5))

8. Determine the eccentricity using the formula e=c/a

e=√(,5)/3

Final Answer

Center: *(0,0), Vertices: *(0,±3), Co-vertices: *(±2,0), Foci: *(0,±√(,5)), Eccentricity: √(,5)/3

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