The line $21x + 5y = k$ touches the hyperbola $7x^2 - 5y^2 = 232$. Then $k =$

$A$ point on the curve $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ is

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola,whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$,where $\alpha > 0$,lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$,then $\alpha^2+\beta$ is equal to:

The eccentricity of the curve $x^2 - y^2 = a^2$ is

For the hyperbola $\frac{x^{2}}{\cos^{2} \alpha} - \frac{y^{2}}{\sin^{2} \alpha} = 1$,which of the following remains fixed when $\alpha$ varies?

If the line $y = 2x + \lambda$ is a tangent to the hyperbola $36x^2 - 25y^2 = 3600$,then $\lambda = $