Chapter 10, Problem 8CT

To determine

The identification of the conic equation $3x^2-xy+2y^2+3y+1=0$ without completing the squares and without rotating axes.

Answer to Problem 8CT

Solution:

The conic equation $3x^2-xy+2y^2+3y+1=0$ is ellipse.

Explanation of Solution

Given information:

The equation $3x^2-xy+2y^2+3y+1=0$

Consider the equation $3x^2-xy+2y^2+3y+1=0$

The general equation of the conic is of the form $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ .

(i) Defines a parabola if $B^2-4AC=0$ .

(ii) Defines a ellipse if $B^2-4AC<0$ and $A\ne B$ .

(iii) Defines a circle if $B^2-4AC<0$ and $A=B$ .

(iv) Defines a hyperbola if $B^2-4AC>0$

Compare the equation $3x^2-xy+2y^2+3y+1=0$ with $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$,

  $A=3,B=-1,C=2,D=0,E=3$, and $F=1$ .

Now, find $B^2-4AC$

  $B^2-4AC={(-1)}^2-4\cdot 3\cdot 2$

  $\Rightarrow B^2-4AC=1-24$

  $\Rightarrow B^2-4AC=-23$

For the given equation, $B^2-4AC<0$ and $A\ne B$

Hence, the conic equation $9x^2-12xy+4y^2+8x+13y=0$ is an ellipse.