Graph (x^2)/25+(y^2)/9=1

Problem

(x^2)/25+(y^2)/9=1

Solution

1. Identify the type of conic section. Since the equation is in the form (x^2)/(a^2)+(y^2)/(b^2)=1 with positive coefficients, it represents an ellipse centered at the origin (0,0)

2. Determine the lengths of the semi-axes. Here, a^2=25 and b^2=9 Taking the square roots gives a=5 and b=3

3. Locate the vertices on the x-axis. Since a=5 is under the x^2 term, the horizontal vertices are at (5,0) and (−5,0)

4. Locate the vertices on the y-axis. Since b=3 is under the y^2 term, the vertical co-vertices are at (0,3) and (0,−3)

5. Calculate the foci using the relation c^2=a^2−b^2

c^2=25−9

c^2=16

c=4

6. Identify the foci coordinates. The foci are located along the major axis (the x-axis) at (4,0) and (−4,0)

7. Sketch the graph by drawing a smooth curve through the four vertices (5,0) (−5,0) (0,3) and (0,−3)

Final Answer

(x^2)/25+(y^2)/9=1* is an ellipse with vertices *(±5,0)* and *(0,±3)

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