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

The locus of a point $P (h, k)$ such that the line $y = hx + k$ is tangent to $4x^2 - 3y^2 = 1$ , is a/an

The equation of the hyperbola in the standard form (with transverse axis along the $x$ -  axis) having the length of the latus rectum = $9$ units and eccentricity = $5/4$ is

The eccentricity of the hyperbola ${x^2} - {y^2} = 25$ is

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$

Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$, then the eccentricity of the hyperbola is