If $2x - ky + 3 = 0$ and $3x - y + 1 = 0$ are conjugate lines with respect to the hyperbola $5x^2 - 6y^2 = 15$,then $k =$

Let $x^2+y^2=16$ be the equation of the auxiliary circle of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and let $(4 \sqrt{2}, 3)$ be a point on the hyperbola. Then,the eccentricity of the hyperbola is

If the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,passing through $(6, 4\sqrt{3})$,satisfies $15(e^2 + 1) = 34e$,then the length of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{2(a^2 + 1)} = 1$ is:

The eccentricity of the hyperbola $4x^2 - 9y^2 = 16$ is

The foci of the hyperbola $9x^2 - 16y^2 = 144$ are

If one of the roots of the equation $x^2 - 5x - 14 = 0$ is the length of the semi-conjugate axis of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the square of the other root is the semi-transverse axis,then the focus of the hyperbola that lies on the positive $x$-axis is