Question 549004
<pre>
4x² - 3xy = 18
 
We write it in the form of the general conic:
 
Ax² + Bxy + Cy² + Dx + Ey + F = 0
 
4x² - 3xy + 0y² + 0x + 0y - 18 = 0
 
So A=4, B=-3, C=0, D=0, E=0, F=-18
 
To find the angle to rotate the graph, we use:
 
tan(2&#415;) = {{{B/(A-C)}}} 
 
tan(2&#415;) = {{{(-3)/(4-0)}}}
 
tan(2&#415;) = {{{-3/4}}}
 
Use the identity for tan(2&#415;)
 
tan(2&#415;) = {{{2tan(theta)/(1-"tan"^2(theta))}}}
 
{{{2tan(theta)/(1-"tan"^2(theta))}}} = {{{-3/4}}}
 
8tan&#415; = -3(1 - tan²&#415;)
 
8tan&#415; = -3 + 3tan²&#415;
 
-3tan²&#415; + 8tan&#415; + 3 = 0
 
3tan²&#415; - 8tan&#415; - 3 = 0
 
That factors:
 
(tan&#415; - 3)(3tan&#415; + 1) = 0
 
tan&#415; - 3 = 0          3tan&#415; + 1 = 0  
                         
    tan&#415; = 3               tan&#415; = {{{-1/3}}}
 
       &#415; = arctan(3)          &#415; = arctan({{{-1/3}}})
                    
   &#415; = 71.56505118°           &#415; = 161.5650512° or 341.5650512° 
       or 251.5650512°
 
Either of those angles will eliminate the xy term when rotated through
them.  We choose &#415; = arctan(3) = 71.56505118°.
 
We draw the triangle:  {{{drawing(400/7,100,-.5,1.5,-.5,3,triangle(0,0,1,0,1,3), locate(.5,0,1),locate(1,1.5,3),locate(.25,.6,theta) )}}} and use the Pythagorean theorem to find the hypotenuse: {{{drawing(400/7,100,-.5,1.5,-.5,3,triangle(0,0,1,0,1,3), locate(.5,0,1),locate(1,1.5,3),locate(-.5,1.5,sqrt(10)),locate(.25,.6,theta) )}}} 
 
 Next we substitute 

x = x'cos&#415;-y'sin&#415; = x'{{{1/sqrt(10)}}}-y'{{{3/sqrt(10)}}} = {{{1/sqrt(10)}}}(x' - 3y') 
y = x'sin&#415;+y'cos&#415; = x'{{{3/sqrt(10)}}}+y'{{{1/sqrt(10)}}} = {{{1/sqrt(10)}}}(3x' + y') 


in

4x² - 3xy = 18
4({{{1/sqrt(10)}}}(x' - 3y'))² - 3({{{1/sqrt(10)}}}(x' - 3y'))({{{1/sqrt(10)}}}(3x' + y')) = 18

4({{{1/sqrt(10)}}})²(x' - 3y')² - 3({{{1/sqrt(10)}}})²(x' - 3y')(3x' + y') = 18

4({{{1/10}}})(x' - 3y')² - 3({{{1/10)}}})(x' - 3y')(3x' + y') = 18

Multiply through by 10

4(x' - 3y')² - 3(x' - 3y')(3x' + y') = 180

4(x'² - 6x'y' + 9y'²) - 3(3x'² - 8x'y' - 3y'²) = 180

4x'² - 24x'y' + 36y'² - 9x'² + 24x'y' + 9y'² = 180
 
-5x'² + 45y'² = 180

Divide through by 180 to get 1 on the right side

{{{(-5)/180}}}x'² + {{{45/180}}}y'² = {{{180/180}}}

{{{"-x'"^2/36}}} + {{{"y'"^2/4}}} = 1

Write the positive term first:

{{{"y'"^2/4}}} - {{{"x'"^2/36}}} = 1

This is a hyperbola of the form {{{y^2/a^2}}} - {{{x^2/b^2}}} = 1
rotated through an angle of &#415; = arctan(3) = 71.56505118°.
a = 2 and b = 6

We will draw the x' and y' axes in green:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),


green(line(-14,-42,14,42),line(-15,5,15,-5)),locate(3.5,10,"x'"),
locate(-10,3.5,"y'")

)}}} 

We draw in the defining rectangle, 6 units out the x' axis in
both directions, and 2 units out the y' axis in both directions. 


{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
green(line(0,6.32455532,-3.794733192,-5.059649257),
line(3.794733192,5.059649257,0,-6.32455532),
line(3.794733192,5.059649257,0,6.32455532),
line(-3.794733192,-5.059649257,0,-6.32455532)),



green(line(-14,-42,14,42),line(-15,5,15,-5)),locate(3.5,10,"x'"),
locate(-10,3.5,"y'")

)}}} 

We draw in the asymptotes as the extended diagonals of the
defining rectangle, one of which turns out to be the y-axis:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
green(line(0,6.32455532,-3.794733192,-5.059649257),
line(3.794733192,5.059649257,0,-6.32455532),
line(3.794733192,5.059649257,0,6.32455532),
line(-3.794733192,-5.059649257,0,-6.32455532),
line(-15,-20,12,16)),



green(line(-14,-42,14,42),line(-15,5,15,-5)),locate(3.5,10,"x'"),
locate(-10,3.5,"y'")

)}}}

And we sketch in the hyperbola:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
green(line(0,6.32455532,-3.794733192,-5.059649257),
line(3.794733192,5.059649257,0,-6.32455532),
line(3.794733192,5.059649257,0,6.32455532),
line(-3.794733192,-5.059649257,0,-6.32455532),
line(-15,-20,12,16)),

graph(400,400,-10,10,-10,10,(18-4x^2)/(-3x)),



green(line(-14,-42,14,42),line(-15,5,15,-5)),locate(3.5,10,"x'"),
locate(-10,3.5,"y'")

)}}}




 
 Edwin</pre>