An ellipse whose one focus is $(4,3)$ passes through $(1,2)$ and equation of tangent at $(1,2)$ is $x+y-3=0$. If the abscissa of centre of ellipse is $7$, find the length of minor axis and eccentricity of the ellipse.
My Attempt:
Let the centre be $(7,k)$
If one focus is $(4,3)$, the other focus is $(10,2k-3)$
Can we say the equation of tangent is $y-k=m(x-7)\pm\sqrt{a^2m^2+b^2}$, where $a,b$ are the lengths of semi-major and minor axis.
From given equation, $m=-1$ and intercept $=3$. So, we'll get one equation in $a,b,k$
Also, sum of focal distances of a point is equal to length of major axis. So, from here we'll get one equation in $a,k$
But we have three variables, so, need three equations.
Not able to proceed ahead.
If the abscissa of center is $7$, then the abscissa of second focus is $10$. The second focus also lies on the line passing through $P=(1,2)$ and through $(2,5)$ (the reflection of the first focus about the normal at $P$). Hence the second focus is $(10,29)$.