Week 3: Rotation of Conics

We looked at the rotation of conics. In an equation it would have Ax^2 +Bxy +cy^2+ Dx + Ey + F=0

There cannot be a Bxy in an equaiton at all.

To get rid of it you first

Step 1: Find the angle with cot2θ=A-C /B 0<θ<90

Step 2: You replace x&y x=x^1 cosθ-y^1 sinθ and y=x^1 sinθ+y^1cosθ

Step 3: Then you use Algebra to simplify

It sounds easy at first but it is so much more.

Here’s some helpful hints:

1+cot^2 θ=csc^2 θ

cos2θ

sin2θ

and

sinθ=√1-cos2θ/2

cosθ=√1+cos2θ/2

Here is one problem below

It is a lot more complicated. Just like the steps you plug in to find the angle and then replace the x and y to find the answer and then simplify. 

Below is another example:

This time, instead of looking for the angle, you put it into the sinθ/cosθ equation and solve. 

Then there will be times when you solve but you just want to find out what shape the equation is. Then you use 

B^2 - 4AC = 0 Parabola

B^2 - 4AC < 0 Ellipse

B^2 - 4AC > 0 Hyperbola

Below is an example and then two problems

This is how you solve rotation of conics

SEE YOU NEXT WEEK!!!