Question

Formulate a strategy for discussing and graphing an equation of the form

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Short answer

Identify the value of A, B, C in the given equation to identify the conic which it represents

Use the discriminant formula

Find the value of $\theta$

Find the value of $\sin \theta and\cos \theta$

Find the value of for x and y

Graph the equation.

Step-by-step solution

Step 1. Given information

Formulate a strategy for discussing and graphing an equation of the form

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Step 2. Identify A, B and C

Given equation$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

First, we have to identify the value of A, B and C in the given equation with which we will be able to identify the conic it represents which we need to graph

Use the discriminant formula $B^2-4AC$to identify the conic.

If $B^2-4AC=0$then the conic is a parabola.

If $B^2-4AC<0$then the conic is an ellipse.

If $B^2-4AC>0$then the conic is a hyperbola.

Step 3. Find value of θ

Use the equation $\mathrm{co}t2\theta =\frac{A-C}{B}$to find the value of $\theta$.

To find $\theta$we can use the trigonometric identities $\frac{1}{\tan 2\theta }$or use the trigonometric ratios like

role="math" localid="1648440430729" $\mathrm{co}t\theta =\frac{x}{y}$also the value of $\cos 2\theta =\frac{x}{r}$where $r=\sqrt{x^2+y^2}$

If $cot2\theta <0$, then role="math" localid="1648440509691" ${90}^0<2\theta <{180}^0$, and ${45}^0<\theta <{90}^0$.

If $cot2\theta \ge 0$, then ${45}^0<2\theta \le {90}^0$, and$0^0<\theta \le {45}^0$.

Step 4. Identify value of sinθ and cosθ

Use the trigonometric ration $cot2\theta =\frac{A-C}{B}$to find the value of $\sin \theta and\cos \theta$or use the equations $\sin \theta =\sqrt{\frac{1-\cos 2\theta }{2}}$and$\cos \theta =\sqrt{\frac{1+\cos 2\theta }{2}}$if the value of$\cos 2\theta$is found.

Step 5. Find x and y

To find the value of x and y, substitute the value of $\sin \theta and\cos \theta$in the equation

role="math" localid="1648441213606" $$\begin{gathered} x=x^{\text{'}}\cos \theta -y^{\text{'}}\sin \theta \\ y=x^{\text{'}}\sin \theta +y^{\text{'}}\cos \theta \end{gathered}$$

Step 6. Substitute the values

In the given equation substitute the values of x and y, from the previous step, on simplifying the terms with xy variables will be canceled.

Graph the equation obtained without $xy$variables.