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
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 $a,b,c$.