Chapter 10, Problem 38RE

To determine

Tofind:the conic without completing the square and rotational axes.

Answer to Problem 38RE

The equation is parabolic equation.

Explanation of Solution

Given:

  ${\text{2x}}^{\text{2}}\text{-y+8x=0}$ .

Concept used:

The general equation of conic:

  ${\text{Ax}}^{\text{2}}{\text{+Bxy+Cy}}^{\text{2}}\text{+Dx+Ey+F=0}$

Defines:

A parabola if ${\text{B}}^{\text{2}}\text{-4AC=0}$ .

An ellipse if ${\text{B}}^{\text{2}}\text{-4AC<0}$ $\text{and}$ $\text{A}\ne \text{B}$ .

A circle if ${\text{B}}^{\text{2}}\text{-4AC<0}$ $\text{and}$ $\text{A=B}$ .

A hyperbola if ${\text{B}}^{\text{2}}\text{-4AC>0}$ .

The general equation of conic if $\text{B=0}$ $\text{and}$ $\text{A and B}$ are not both zero.

  ${\text{Ax}}^{\text{2}}{\text{+Cy}}^{\text{2}}\text{+Dx+Ey+F=0}$

Defines:

A parabola if $\text{AC=0}$ .

Here either $\text{A=0 or B=0}$ but not both.

An ellipse if $\text{AC>0}$ $\text{and}$ $\text{A}\ne \text{B}$ .

That is $\text{A and C}$ have same sign, but $\text{A}\ne \text{C}$ .

A circle if $\text{AC>0}$ and $\text{A=C}$

That is $\text{A=C}$ .

A hyperbola if $\text{AC<0}$ .

That is $\text{A}$ $\text{and C}$ have opposite signs.

Calculation:

Here $\text{A=0}$ , $\text{B=0}$ and $\text{C=1}$ .

According to the formula:

The general equation of conic if $\text{B=0}$ $\text{and}$ $\text{A and B}$ are not both zero.

  ${\text{Ax}}^{\text{2}}{\text{+Cy}}^{\text{2}}\text{+Dx+Ey+F=0}$ .

Defines:

A parabola if $\text{AC=0}$ .

Either $\text{A=0 or B=0}$ but not both.

  ${\text{2x}}^{\text{2}}\text{-y+8x=0}$ by comparing with general equation:

Here $\text{A=2}$ , $\text{B=0}$

  $\text{C=0}$ .

By putting the value of $\text{A,B and C}$ .

It satisfies the equation of parabola

Hence the equation is parabolic equation.