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Chapter 11: Problem 7

What is the equation of the standard ellipse with vertices at \((\pm a, 0)\) and foci at \((\pm c, 0) ?\)

Short Answer

Expert verified
Answer: The equation of a standard ellipse with vertices at $(\pm a, 0)$ and foci at $(\pm c, 0)$ is $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{(a^2-c^2)}=1$.

Step by step solution

01

Recall the standard equation of an ellipse

The standard equation of an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\) is given by the expression: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]
02

Identify the given information

The vertices of the ellipse are given as \((\pm a, 0)\) and the foci as \((\pm c, 0)\). From this information, we know that the ellipse is horizontal, and the semi-major axis (\(a\)) is given directly.
03

Use the relationship between \(a\), \(b\), and \(c\)

The relationship between the semi-major axis \(a\), semi-minor axis \(b\), and the distance between the center and the foci \(c\) is given by: \[a^2 - b^2 = c^2\]
04

Solve for \(b^2\)

From the given information, we have \(c^2\). We can rewrite the relationship found in step 3, solving for \(b^2\): \[b^2 = a^2 - c^2\]
05

Substitute values in the standard equation of the ellipse

From the analysis above, we can now substitute the values of \(a^2\) and \(b^2\) in the standard equation of the ellipse to obtain the final equation: \[\frac{x^2}{a^2}+\frac{y^2}{(a^2-c^2)}=1\] There we have it, the equation of the standard ellipse with vertices at \((\pm a, 0)\) and foci at \((\pm c, 0)\) is given by: \[\frac{x^2}{a^2}+\frac{y^2}{(a^2-c^2)}=1\]

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Most popular questions from this chapter

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