Chapter 11, Problem 58RE

a.

To determine

To find: Whether the graph of the equation is a parabola, an ellipse or a hyperbola.

Answer to Problem 58RE

The graph of the equation is not a parabola, neither ellipse nor a hyperbola

Explanation of Solution

Given information:

The equation of the graph is $9x^2+24xy+16y^2=25$

Concept used:

The general equation of the form

  $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

The discremanant is

  $\Delta =\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$

If $\Delta =0$ then it represents degenerate conic

  $h^2-ab>0$ ; represents two distant real lines

  $h^2-ab=0$ ; represents non - lines

  $h^2-ab<0$ ; represents parallel lines

If $\Delta \ne 0$ then it generate non − degenerate conic

  $h^2-ab>0$ ; represents hyperbola

  $h^2-ab=0$ ; represents parabola

  $h^2-ab<0$ ; represents ellipse

Calculation:

The given equation is $9x^2+24xy+16y^2=25$

Comparing with general equation

  $\begin{array}{l}a=9\\ b=16\\ h=12\\ c=-25\\ g=0\\ f=0\end{array}$

Find $△$ using these values

  $△=\left|\begin{array}{ccc}9& 12& 0\\ 12& 16& 0\\ 0& 0& -25\end{array}\right|=0$

Find $h^2-ab$

  $h^2-ab={\left(12\right)}^2-\left(9\right)\left(16\right)=0$

Conclusion:

Therefore the equation of the graph represents the parallel lines.

b.

To determine

To eliminate: $xy$ term in the given equation.

Answer to Problem 58RE

Required equation after eliminating $xy$ term is

  $y^2-1=0$

Explanation of Solution

Given information:

The equation of the graph is

  $9x^2+24xy+16y^2-25=0$

Concept used:

Let $\theta$ be the angle of rotation of axis then

  $\mathrm{Tan}2\theta =\left(\frac{2h}{a-b}\right)$

Using this find $\sin \theta$ and $\cos \theta$

The new axis will be

  $\begin{array}{l}x=x\text{'}\cos \theta -y\text{'}\sin \theta \\ y=x\text{'}\sin \theta +y\text{'}\cos \theta \end{array}$

Substitute x & y in the given equation t obtain the required equation.

Calculation:

Comparing the given equation

  $9x^2+24xy+16y^2-25=0$

with the general equation $a{x}^2+2hxy+b{y}^2+2gx+c=0$

  $\begin{array}{l}a=9\\ b=16\\ h=12\\ c=-25\\ g=0\\ f=0\end{array}$

Substitute the values find $\sin \theta$ and $\cos \theta$

  $\mathrm{Tan}2\theta =\frac{2h}{a-b}=\frac{24}{7}$

On simplifying

  $\cos \theta =\frac{4}{5}$ and $\sin \theta =\frac{3}{5}$

The new axis are

  $\begin{array}{l}x=x\text{'}\cos \theta -y\text{'}\sin \theta =\frac{4x\text{'}}{5}+\frac{3y\text{'}}{5}\\ y=x\text{'}\sin \theta +y\text{'}\cos \theta =-\frac{3x\text{'}}{5}+\frac{4y\text{'}}{5}\end{array}$

Substitute in the given equation

  $\begin{array}{l}9x^2+24xy+16y^2-25=0\\ \Rightarrow 9{\left(\frac{4x\text{'}}{5}+\frac{3y\text{'}}{5}\right)}^2+24\left(\frac{4x\text{'}}{5}+\frac{3y\text{'}}{5}\right)\left(-\frac{3x\text{'}}{5}+\frac{4y\text{'}}{5}\right)+16{\left(-\frac{3x\text{'}}{5}+\frac{4y\text{'}}{5}\right)}^2-25=0\end{array}$

On Simplifying

  $y^2-1=0$

Conclusion:

Therefore the required equation after eliminating the $xy$ term is

  $y^2-1=0$

c.

To determine

To sketch the graph of the given equation

Answer to Problem 58RE

From the graph it can be observed that the given equation represents an parallel lines.

Explanation of Solution

Given information:

Given equation of graph is $9x^2+24xy+16y^2-25=0$

Graph:

Sketch the graph using graphing utility of the polar coordinates.

The graph is obtained as:

  

Interpretation:

From the above graph it can be observed that the given equation represents an parallel lines.