<s>[INST] An ellipse has foci at $(1, -1)$ and $(2, -1)$ and tangent $x+y-5=0$. Find the point where the tangent touches the ellipse. 
An ellipse has foci at $(1, -1)$ and $(2, -1)$ and tangent $x+y-5=0$. Find the point where the tangent touches the ellipse.

Here is a procedure how to do it analiticaly.

*

*If $T(x_0,y_0)$ is a touching point, then $x_0+y_0=5$

*The equation of ellipse is $${(x_0-{3\over 2})^2\over a^2} +{(y_0+1)^2 \over b^2}=1$$

*Since $2e=1$ we have $a^2-b^2 = {1\over 4}$

*Since the slope of tangent is $-1$ we have $${2(x_0-{3\over 2})\over a^2} -{2(y_0+1)\over b^2}=0$$
And now we have to solve this tedious system. How to do it more geometrical?
 [/INST] We have the coordinates of any point on the ellipse centered at the origin $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in parametric form as $P(a\cos\theta,b\sin\theta)$, where $\theta$ is a parameter. Now, by using basic calculus, we get the equation of tangent to the ellipse at this point as $\frac{x}{a^2}a\cos\theta+\frac{y}{b^2}b\sin\theta=1$.
Now, by the concept of shifting of origin, we can obtain similar results for the ellipse given in the question.
$\Rightarrow$ Equation of ellipse: $$\frac{(x-\frac32)^2}{a^2}+\frac{(y+1)^2}{b^2}=1$$
$\Rightarrow$ Coordinates of any point on the ellipse in parametric form: $Q(a\cos\theta+\frac32,b\sin\theta-1)$
$\Rightarrow$ Equation of tangent to the ellipse at this point: $$\frac{(x-\frac32)}{a^2}a\cos\theta+\frac{(y+1)}{b^2}b\sin\theta=1$$
$$\Rightarrow \frac xa \cos\theta+\frac yb \sin\theta=\frac{3\cos\theta}{2a}-\frac{\sin\theta}{b}+1$$
Comparing this with the given equation of tangent $x+y=5$, we get $$\frac{\cos\theta}{a}=\frac{\sin\theta}{b}=\frac{\frac{3\cos\theta}{2a}-\frac{\sin\theta}{b}+1}{5}\ldots(1)$$
Also, we have $$a^2-b^2=\frac14\ldots(2)$$
Now, we can assume $\frac{\cos\theta}{a}$ and $\frac{\sin\theta}{b}$ as two single variables and solve for them to get $$\Rightarrow\frac{\cos\theta}{a}=\frac{\sin\theta}{b}=\frac29\ldots(3)$$
$(2)$ and $(3)$ can easily be solved and we get $\cos\theta=\sqrt\frac{41}{81}$ and $ \sin\theta=\sqrt\frac{40}{81}$.
Now you can easily obtain the coordinates of $Q$ by substituting $a$ and $b$ in terms of $\cos\theta$ and $\sin\theta$.
Therefore, $$\Rightarrow Q=(\frac{34}{9},\frac{11}{9})$$
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