Find the Properties (x^2)/25+(y^2)/64=1

Problem

(x2)/25+(y2)/64=1

Solution

1. Identify the type of conic section by observing the equation is in the form (x2)/(b2)+(y2)/(a2)=1 where a2>b2 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=64⇒a=8

b2=25⇒b=5

4. Find the vertices and co-vertices using the values of a and b relative to the center.

Vertices:(0,±a)=(0,±8)

Co-vertices:(±b,0)=(±5,0)

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

c2=64−25=39

c=√(,39)

6. Identify the foci located along the major axis (y-axis) at a distance c from the center.

Foci:(0,±√(,39))

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

e=√(,39)/8

Final Answer

Type: Vertical Ellipse, Center: *(0,0), Vertices: *(0,±8), Foci: *(0,±√(,39)), Eccentricity: √(,39)/8

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