On the ellipse $4x^2 + 9y^2 = 1$,the points at which the tangents are parallel to the line $8x = 9y$ are

Let the foci of the ellipse $\frac{x^{2}}{9}+y^{2}=1$ subtend a right angle at a point $P$. Then, the locus of $P$ is

Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis,respectively. Let $S$ be the circle $x^2+(y-1)^2=2$. The straight line $x+y=3$ touches the curves $S, E_1$ and $E_2$ at $P, Q$ and $R$,respectively. Suppose that $PQ=PR=\frac{2 \sqrt{2}}{3}$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$,respectively,then the correct expression$(s)$ is(are):
$(A) e_1^2+e_2^2=\frac{43}{40}$
$(B) e_1 e_2=\frac{\sqrt{7}}{2 \sqrt{10}}$
$(C) |e_1^2-e_2^2|=\frac{5}{8}$
$(D) e_1 e_2=\frac{\sqrt{3}}{4}$

The eccentricity of the conic $16x^2 + 7y^2 = 112$ is

Let a tangent to the curve $9x^2 + 16y^2 = 144$ intersect the coordinate axes at the points $A$ and $B$. Then,the minimum length of the line segment $AB$ is $.........$

The point $(1,3)$ with respect to the ellipse $4x^2+9y^2-16x-54y+61=0$ lies