Identify nondegenerate conic sections given their general form equations.
Write equations of rotated conics in standard form.
Identify conics without rotating axes.
A plaza designer tests a tilted spotlight that projects a conic-shaped pool of light onto the ground. With ground axes aligned to nearby paving joints, measurements of the light boundary fit the equation [latex]8x^{2}-12xy+17y^{2}=20.[/latex]
Identify the conic directly from the general form (without rotating axes).
Write the rotated standard form by eliminating the [latex]xy[/latex] term. For rotation angle [latex]\theta[/latex], use [latex]\cot(2\theta)=\dfrac{A-C}{B}[/latex].
General form: [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[/latex] with [latex]A=8,;B=-12,;C=17[/latex]. Compute the discriminant [latex]B^{2}-4AC[/latex].[latex]\\[/latex][latex]\begin{align} B^{2}-4AC&=(-12)^{2}-4(8)(17)\\ &=144-544\\ &=-400<0 \end{align}[/latex]
[latex]\\[/latex]
Because [latex]B^{2}-4AC<0[/latex], the curve is an ellipse.
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Standard form relative to the rotated axes is [latex]\dfrac{x'^2}{4}+\dfrac{y'^2}{1}=1,[/latex] an ellipse with semi-axes [latex]a=2[/latex] and [latex]b=1[/latex] (meters, in this context). The rotation angle satisfies [latex]\cot(2\theta)=3/4[/latex], so [latex]\theta\approx 18.435^\circ[/latex].
Use the discriminant [latex]B^{2}-4AC[/latex] to classify without rotating. • To remove [latex]xy[/latex], rotate by [latex]\cot(2\theta)=\dfrac{A-C}{B}[/latex] and substitute [latex]x=x'\cos\theta-y'\sin\theta,;y=x'\sin\theta+y'\cos\theta[/latex]. • Write the result in standard form in the [latex]x',y'[/latex] system.
A survey drone maps a reflective sculpture; a cross-section relative to chosen ground axes is modeled by [latex]5x^{2}+6xy+5y^{2}=50.[/latex]
a) Identify the conic without rotating axes.
b) Find a rotation that eliminates [latex]xy[/latex] and write the standard form in the [latex]x',y'[/latex] system.