Find the Properties (x^2)/16+(y^2)/9=1

Problem

(x2)/16+(y2)/9=1

Solution

1. Identify the type of conic section. Since the equation is in the form (x2)/(a2)+(y2)/(b2)=1 with a2>b2 it is a horizontal ellipse centered at the origin (0,0)

2. Determine the values of a and b We have a2=16 which gives a=4 and b2=9 which gives b=3

3. Calculate the distance to the foci c using the relation c2=a2−b2

c2=16−9

c2=7

c=√(,7)

4. Find the vertices and co-vertices. The vertices are at (±a,0) which are (±4,0) The co-vertices are at (0,±b) which are (0,±3)

5. Locate the foci. The foci are at (±c,0) which are (±√(,7),0)

6. Compute the eccentricity e using the formula e=c/a

e=√(,7)/4

Final Answer

Center: *(0,0), Vertices: *(±4,0), Co-vertices: *(0,±3), Foci: *(±√(,7),0), Eccentricity: √(,7)/4

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