12.4 Rotation of axes  (Page 5/8)

What effect does the $xy$ term have on the graph of a conic section?

If the equation of a conic section is written in the form $A{x}^2+B{y}^2+Cx+Dy+E=0$ and $AB=0,$ what can we conclude?

If the equation of a conic section is written in the form $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$ and $B^2-4AC>0,$ what can we conclude?

Given the equation $a{x}^2+4x+3y^2-12=0,$ what can we conclude if $a>0?$

For the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$ the value of $\theta$ that satisfies $\cot \left(2\theta \right)=\frac{A-C}{B}$ gives us what information?

$9x^2+4y^2+72x+36y-500=0$

$x^2-10x+4y-10=0$

$2x^2-2y^2+4x-6y-2=0$

$4x^2-y^2+8x-1=0$

$4y^2-5x+9y+1=0$

$2x^2+3y^2-8x-12y+2=0$

$4x^2+9xy+4y^2-36y-125=0$

$3x^2+6xy+3y^2-36y-125=0$

$-3x^2+3\sqrt{3}xy-4y^2+9=0$

$2x^2+4\sqrt{3}xy+6y^2-6x-3=0$

$-x^2+4\sqrt{2}xy+2y^2-2y+1=0$

$8x^2+4\sqrt{2}xy+4y^2-10x+1=0$

$3x^2+xy+3y^2-5=0,\theta =\mathrm{45°}$

$4x^2-xy+4y^2-2=0,\theta =\mathrm{45°}$

$2x^2+8xy-1=0,\theta =\mathrm{30°}$

$-2x^2+8xy+1=0,\theta =\mathrm{45°}$

$4x^2+\sqrt{2}xy+4y^2+y+2=0,\theta =\mathrm{45°}$

$x^2+3\sqrt{3}xy+4y^2+y-2=0$

$4x^2+2\sqrt{3}xy+6y^2+y-2=0$

$9x^2-3\sqrt{3}xy+6y^2+4y-3=0$

$\mathrm{-3}{x}^2-\sqrt{3}xy-2y^2-x=0$

$16x^2+24xy+9y^2+6x-6y+2=0$

$x^2+4xy+4y^2+3x-2=0$

$x^2+4xy+y^2-2x+1=0$

$4x^2-2\sqrt{3}xy+6y^2-1=0$

$y=-x^2,\theta =-{45}^{\circ }$

$x=y^2,\theta ={45}^{\circ }$

$\frac{x^2}{4}+\frac{y^2}{1}=1,\theta ={45}^{\circ }$

$\frac{y^2}{16}+\frac{x^2}{9}=1,\theta ={45}^{\circ }$

$y^2-x^2=1,\theta ={45}^{\circ }$

$y=\frac{x^2}{2},\theta ={30}^{\circ }$

$x={\left(y-1\right)}^2,\theta ={30}^{\circ }$

$\frac{x^2}{9}+\frac{y^2}{4}=1,\theta ={30}^{\circ }$

$xy=9$

$x^2+10xy+y^2-6=0$

$x^2-10xy+y^2-24=0$

$4x^2-3\sqrt{3}xy+y^2-22=0$

$6x^2+2\sqrt{3}xy+4y^2-21=0$

$11x^2+10\sqrt{3}xy+y^2-64=0$

$21x^2+2\sqrt{3}xy+19y^2-18=0$

$16x^2+24xy+9y^2-130x+90y=0$

$16x^2+24xy+9y^2-60x+80y=0$

$13x^2-6\sqrt{3}xy+7y^2-16=0$

$4x^2-4xy+y^2-8\sqrt{5}x-16\sqrt{5}y=0$

$6x^2-5\sqrt{3}xy+y^2+10x-12y=0$

$6x^2-5xy+6y^2+20x-y=0$

$6x^2-8\sqrt{3}xy+14y^2+10x-3y=0$

$4x^2+6\sqrt{3}xy+10y^2+20x-40y=0$

$8x^2+3xy+4y^2+2x-4=0$

$16x^2+24xy+9y^2+20x-44y=0$

Given $4x^2+kxy+16y^2+8x+24y-48=0,$ find $k$ for the graph to be a parabola.

Given $2x^2+kxy+12y^2+10x-16y+28=0,$ find $k$ for the graph to be an ellipse.

Given $3x^2+kxy+4y^2-6x+20y+128=0,$ find $k$ for the graph to be a hyperbola.

Given $k{x}^2+8xy+8y^2-12x+16y+18=0,$ find $k$ for the graph to be a parabola.

Given $6x^2+12xy+k{y}^2+16x+10y+4=0,$ find $k$ for the graph to be an ellipse.