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

If $5x^2 + \lambda y^2 = 20$ represents a rectangular hyperbola,then $\lambda$ equals:

The point $P(-2 \sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having eccentricity $e = \frac{\sqrt{5}}{2}$. If the tangent and normal at $P$ to the hyperbola intersect its conjugate axis at the points $Q$ and $R$ respectively,then $QR$ is equal to :

If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2$ in four points $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4)$,then:

The locus of the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = \lambda$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{\lambda}$ (where $\lambda$ is a parameter) is:

The eccentricity of the hyperbola conjugate to ${x^2} - 3{y^2} = 2x + 8$ is