Graph (x^2)/64+(y^2)/81=1

Problem

(x2)/64+(y2)/81=1

Solution

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

2. Determine the values of a2 and b2 Here, a2=64 and b2=81

3. Calculate the lengths of the semi-axes by taking the square roots.

a=√(,64)=8

b=√(,81)=9

4. Locate the vertices and co-vertices. Since b>a the major axis is vertical. The vertices are at (0,±9) and the co-vertices are at (±8,0)

5. Find the foci using the relationship c2=|a2−b2|

c2=81−64=17

c=√(,17)≈4.12

The foci are located at (0,±√(,17))

6. Sketch the graph by plotting the center (0,0) the vertices (0,9) and (0,−9) and the co-vertices (8,0) and (−8,0) then drawing a smooth curve through these points.

Final Answer

(x2)/64+(y2)/81=1* is an ellipse with vertices *(0,±9)* and co-vertices *(±8,0)

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