Question
Write the equation of the ellipse centered at the origin with a vertex at (0,6)
and a co-vertex at (3,0).
and a co-vertex at (3,0).
Answers
GPT-4o mini
The standard form of the equation of an ellipse centered at the origin is given by:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
In this case, the vertex at (0,6) indicates that the semi-major axis \(a\) is equal to 6. Thus, we have:
\[
a = 6 \quad \Rightarrow \quad a^2 = 36.
\]
The co-vertex at (3,0) indicates that the semi-minor axis \(b\) is equal to 3. Therefore:
\[
b = 3 \quad \Rightarrow \quad b^2 = 9.
\]
Since the major axis is vertical (the y-coordinates are larger), we will use the vertical form of the ellipse's equation:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1.
\]
Substituting in the values of \(a^2\) and \(b^2\):
\[
\frac{x^2}{9} + \frac{y^2}{36} = 1.
\]
Thus, the equation of the ellipse is:
\[
\frac{x^2}{9} + \frac{y^2}{36} = 1.
\]
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
In this case, the vertex at (0,6) indicates that the semi-major axis \(a\) is equal to 6. Thus, we have:
\[
a = 6 \quad \Rightarrow \quad a^2 = 36.
\]
The co-vertex at (3,0) indicates that the semi-minor axis \(b\) is equal to 3. Therefore:
\[
b = 3 \quad \Rightarrow \quad b^2 = 9.
\]
Since the major axis is vertical (the y-coordinates are larger), we will use the vertical form of the ellipse's equation:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1.
\]
Substituting in the values of \(a^2\) and \(b^2\):
\[
\frac{x^2}{9} + \frac{y^2}{36} = 1.
\]
Thus, the equation of the ellipse is:
\[
\frac{x^2}{9} + \frac{y^2}{36} = 1.
\]