Question

write the equation of the ellipse that has its center at the origin and a focus at (0,3); vertex at (0,6) question help: video ebook

Explanation:

Step1: Determine the major axis direction

Since the focus \((0,3)\) and vertex \((0,6)\) have the same \(x -\)coordinate, the major axis is vertical. The standard form of the ellipse equation with center \((h,k)=(0,0)\) (origin) and vertical major axis is \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\), where \(a\) is the semi - major axis and \(b\) is the semi - minor axis.

Step2: Find the value of \(a\)

The distance from the center \((0,0)\) to the vertex \((0,6)\) gives \(a\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), for \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,6)\), \(a=\vert6 - 0\vert=6\)

Step3: Find the value of \(c\)

The distance from the center \((0,0)\) to the focus \((0,3)\) gives \(c\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), for \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(0,3)\), \(c=\vert3 - 0\vert = 3\)

Step4: Find the value of \(b\)

We know the relationship \(c^{2}=a^{2}-b^{2}\). Substituting \(a = 6\) and \(c = 3\) into the formula:

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Step5: Write the equation of the ellipse

Substitute \(a^{2}=36\) and \(b^{2}=27\) into the standard form \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\)

$$ \frac{x^{2}}{27}+\frac{y^{2}}{36}=1 $$

Answer:

\(\frac{x^{2}}{27}+\frac{y^{2}}{36}=1\)