Rotation of axes for an ellipse Use rotation of axes to write 21x² - 10√3xy + 31y² = 144 in the standard form of an ellipse and sketch its graph.

Question

Rotation of axes for an ellipse

Use rotation of axes to write [latex] 21 x^{2}-10 \sqrt{3} x y+31 y^{2}=144[/latex] in the standard form of an ellipse and sketch its graph.

Accepted Answer

We have [latex] A=21, B=-10 \sqrt{3}[/latex] , and C = 31. Use

[latex] \cot 2 heta=\frac{A-C}{B}[/latex]

to get [latex] \cot 2 heta=1 / \sqrt{3}, 2 heta=\pi / 3[/latex], and [latex] heta=\pi / 6[/latex]. Now find x and y in terms of x’ and y’:

[latex] \begin{aligned} x &=x^{\prime} \cos \frac{\pi}{6}-y^{\prime} \sin \frac{\pi}{6} & y &=x^{\prime} \sin \frac{\pi}{6}+y^{\prime} \cos \frac{\pi}{6} \ &=\frac{\sqrt{3} x^{\prime}-y^{\prime}}{2} & &=\frac{x^{\prime}+\sqrt{3} y^{\prime}}{2} \end{aligned}[/latex]

Substitute these expressions for x and y into [latex] 21 x^{2}-10 \sqrt{3} x y+31 y^{2}=144[/latex]:

[latex] \begin{array}{r} 21\left(\frac{\sqrt{3} x^{\prime}-y^{\prime}}{2} ight)^{2}-10 \sqrt{3}\left(\frac{\sqrt{3} x^{\prime}-y^{\prime}}{2} ight)\left(\frac{x^{\prime}+\sqrt{3} y^{\prime}}{2} ight) \ +31\left(\frac{x^{\prime}+\sqrt{3} y^{\prime}}{2} ight)^{2}=144 \end{array}[/latex]

Simplifying this equation yields

[latex] \frac{\left(x^{\prime} ight)^{2}}{9}+\frac{\left(y^{\prime} ight)^{2}}{4}=1[/latex]

which is the equation for an ellipse centered at the origin with vertices (±3, 0) and y’-intercepts at (0, ±2), as shown in Fig. 10.46.

Question 10.4.6: Rotation of axes for an ellipse Use rotation of axes to writ...

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