# Find the equation in normal Cartesian Coordinates for the following Conic section

#### lolhelpme

Hi Please help me i really need help and I'm confused and lost and desperately need help with this. Thank you

#### Attachments

• 19.5 KB Views: 5

#### johnsomeone

The trick is to first write down the ellipse's equation *prior to* that 45 degree rotation. Only then are you ready to see what the equation is after you've rotated it.

So, pretending for a moment that there's no 45 degree rotation there, can you write down the equation of the ellipse?

#### lolhelpme

Is it 13x^2 - 10xy + 13y^2 = 72 ?

#### johnsomeone

A quick head calculation (I could've made a mistake too) tells me yes, you're equation is correct, because the 4 points you know all satisfy that equation for an ellipse.

(Note: $$\displaystyle B^2 - 4AC = (-10)^2 - 4(13)(13) = 100 - 4(13)^2 < 0$$, so that is the equation for an ellipse.)

If you got it by "plugging in", that's correct, but I assume you're expected to know how to do it by rotations.

Did you get it by rotating $$\displaystyle \frac{x^2}{3^2} + \frac{y^2}{2^2} = 1$$ by 45 degrees? Or by plugging the 4 known points into $$\displaystyle Ax^2 + Bxy + Cy^2 = K$$?

Last edited:

#### lolhelpme

yes by rotating x^2/3^2 + y^2/2^2 = 1 by 45 degrees