Chapter 8.6, Problem 45E

Solution

To explain: How can the second degree equation classify.

Interpretation:

The general second degree equation of any conic is:

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+f=0$

Now, when $B=0,$ the equation becomes

  $A{x}^2+C{y}^2+Dx+Ey+f=0$

Here, if the coefficients of $x^2$ and $y^2$ are same and non-zero, then the equation represents a circle.

And, if the coefficients of $x^2$ and $y^2$ are positive and different, then the equation represents an ellipse.

And, if either of the coefficients of $x^2$ and $y^2$ is negative and the other is positive, then the equation represents a hyperbola.

And, if either of the coefficients of either $x^2$ and $y^2$ is zero, then the equation represents a parabola.

And, if both of the coefficients of either $x^2$ and $y^2$ are zero, then the equation represents a line.