The equations of the tangents to the ellipse $9x^2 + 16y^2 = 144$ which pass through the point $(2, 3)$ are

The line $y=x+\lambda$ is tangent to the ellipse $2x^{2}+3y^{2}=1$. Then, $\lambda$ is

Let $S(1,0)$ and $S^{\prime}(0,1)$ be the foci of an ellipse such that $SP+S^{\prime} P=2$ for any point $P$ on the ellipse. If $A(x_1, y_1)$ and $A^{\prime}(x_2, y_2)$ are the end points of the major axis of this ellipse,then $x_1+x_2=$

$A$ line passing through the endpoint of the major axis $A$ and the endpoint of the minor axis $B$ of an ellipse $\frac{x^2}{9} + y^2 = 1$ touches its auxiliary circle at point $M$. Find the area of the triangle with vertices $A, M,$ and the origin $O$.

An ellipse passing through $(4 \sqrt{2}, 2 \sqrt{6})$ has foci at $(-4, 0)$ and $(4, 0)$. Then,its eccentricity is

Find the locus of a point such that the sum of its distances from the points $(0, 2)$ and $(0, -2)$ is $6$.