Find the equation of the ellipse with given foci and $a$

Question

I have to find the equation of the elipse with foci:

$$(-1,-1),(1,1)$$ and $a = 3$

I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation and rotation. Could somebody help me?

I think I need to rotate it by $45$ degrees

Answer 1

• the center of this ellipse is at origin $(0, 0)$
• let $c$ the distance from center to focal $c = \sqrt{1^2+1^2}$:

$b^2=a^2-c^2$, knowing that $c^2=2$ ==> $b^2=a^2-2$

*thus, the equation of the ellipse is:

$$\frac{x}{a^2}+\frac{y}{b^2}=1$$

and by substituting values :

$$\frac{x}{9}+\frac{y}{7}=1$$

• Now let's rotate by $\theta = \frac{\pi}{4}$:

$x' = x\cos\theta-y\sin\theta$

$y' = x\sin\theta+y\cos\theta$

Solving the above you end up with :

$x=\frac{y'+x'}{\sqrt2}$

$y=\frac{y'-x'}{\sqrt2}$

• substitute in the obtained equation will lead to the ellipse equation:

$$\frac{(x+y)^2}{18}+\frac{(y-x)^2}{14}=1$$