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I have this problem here which i just can't seem to solve:

Find the equation of the ellipse which is tangent to the line $y=-x+3$ and touches said line at the point $P(1,y).$ The ellipse has center in $O(0,0)$ and the major axis is parallel to the x or y axis.

I can't use derivatives to solve it.

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  • $\begingroup$ Welcome to Maths SX!There is no single ellipse which satisfies these conditions! $\endgroup$ – Bernard Jun 12 '18 at 10:29
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    $\begingroup$ Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. $\endgroup$ – José Carlos Santos Jun 12 '18 at 10:31
  • $\begingroup$ Sorry, forgot to mention it has center in O. $\endgroup$ – Luca Giovanni Jun 12 '18 at 10:32
  • $\begingroup$ And you probably forgot to mention that the major axis is parallel either with the x-axis or with the y-axis, right? $\endgroup$ – Reinhard Meier Jun 12 '18 at 10:45
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    $\begingroup$ Hint: Try to find the intersection between ellipse and line. You get a quadratic equation. If this quadratic equation has exactly one solution, you have a point where the line is tangent to the ellipse. $\endgroup$ – Reinhard Meier Jun 12 '18 at 10:49
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Hints:

A conic is centred at origin if and only if its equation is $$ ax^2+2bxy+cy^2=1. $$ This conic is an ellipse if and only if it has no real asymptotes, i.e. ifthe quadratic polynomial $at^2+2bt+c$ has no real root. This means that $$b^2<ac.$$

The conic is tangent to the line $y=-x+3$ if and only if the quadratic equation obtained by elimination of $y$ between both equations has a double root, i.e. its discriminant is $0$.

Furthermore, it is tangent to the line at the given point (with abscissa $1$) if the double root is $1$.

All this results in a system of a linear and a quadratic equations with a constraint, in the coefficients $a,b,c$.

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