Let $e$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

If $a=5, b=4$ and the equation of the normal drawn at one end of the latus rectum that lies in the first quadrant is $l x+m y=27$ then $l+m=$

If the perpendicular distance from the focus of an ellipse $\frac{x^2}{9}+\frac{y^2}{b^2}=1(b<3)$ to its corresponding directrix is $\frac{4}{\sqrt{5}}$, then the slope of the tangent to this ellipse drawn at $\left(\frac{3}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$ is

The length of the chord of the ellipse $\frac{x^2}{4}+y^2=1$ formed on the line $y=x+1$ is

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through $P$ to the major axis meet its auxiliary circle at $Q$. If the normals drawn at $P$ and $Q$ to the ellipse and the auxiliary circle respectively meet in $R$, then the equation of the locus of $R$ is