Question

Replace the polar equation with equivalent Cartesian equations. \frac{x^{2}}{9}+\frac{y^{2}}{4}=1

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asked 2021-06-07
Replace the polar equation with equivalent Cartesian equations. \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)

Answers (1)

2021-06-08
Step 1
We have given a cartesian equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\). The equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is a standard form of ellipse
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and generally, we take polar coordinates x as a \(\cos 0\).
Here, a=3. So set \(x=3 \cos 0\). Now, substitute \(x=3 \cos 0\) in \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)
\(\frac{(3\cos 0)^{2}}{9}+\frac{y^{2}}{4}=1\)
\(\frac{9\cos^{2}0}{9}+\frac{y^{2}}{4}=1\)
Step 2
Use the identity \(1-\cos^{2}0=\sin^{2}0\) to simplify it further.
\(\cos^{2}0+\frac{y^{2}}{4}=1\)
\(\frac{y^{2}}{4}=1-\cos^{2}0\)
\(=\sin^{2}0\)
\(y^{2}=4\sin^{2}0\)
Take square root on both sides to find y. Consider only positive sign.
\(\sqrt{y^{2}}=\pm \sqrt{4\sin^{2}0}\)
\(y=\pm 2\sin 0\)
\(y=2\sin 0\) [Take only (+) sign]
Hence, the polar equation is \(x=3 \cos 0, y=2 \sin 0\).
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