Chapter 10, Problem 84RE

To determine

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

Answer to Problem 84RE

  $\boxed{A\text{'}x+B\text{'}y+C\text{'}}$

Explanation of Solution

Given information:

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

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Calculation:

The given equation is conic form of cylinder,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Now we have to find the conic section and sketch its graph. We need to generate perfect square of $x$ and $y$ on one side of the equation,

  ${\left(x-a\right)}^2+{\left(y-b\right)}^2=\frac{{\left(A\text{'}x+B\text{'}y+C\text{'}\right)}^2}{A{\text{'}}^2+B{\text{'}}^2}$

This equation is a locus of the point equidistance of the point given by coordinates $\left(a,b\right)$ and the line.

  $A\text{'}x+B\text{'}y+C\text{'}$

The line called directrix in case of ellipse and hyperbola.

Hence, equation is $\boxed{A\text{'}x+B\text{'}y+C\text{'}}$