The equation of the tangent to the hyperbola $2{x^2} - 3{y^2} = 6$ which is parallel to the line $y = 3x + 4$, is

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is $................$.

A square $ABCD$ has all its vertices on the curve $x ^{2} y ^{2}=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is

Locus of foot of normal drawn from any focus to variable tangent of hyperbola $4x^2-9y^2-8x- 18y = 41$ will be

If $e$ and $e’$ are the eccentricities of the ellipse $5{x^2} + 9{y^2} = 45$ and the hyperbola $5{x^2} - 4{y^2} = 45$ respectively, then $ee' = $

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is