The locus of the point of intersection of lines $(x + y)t = a$ and $x - y = at$,where $t$ is the parameter,is

The line $2x + y = 1$ is tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis,then the eccentricity of the hyperbola is

If the line $3x - my + 5 = 0$ is a tangent to the hyperbola $3x^2 - 4y^2 = 300$,then the square of the $Y$-intercept made by this tangent line is:

Let $a>0, b>0$. Let $e$ and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Let $e^{\prime}$ and $\ell^{\prime}$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^{2}=\frac{11}{14} \ell$ and $(e^{\prime})^{2}=\frac{11}{8} \ell^{\prime}$,then the value of $77a+44b$ is equal to

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$.

If $\theta$ is the acute angle between the tangents drawn from the point $(2,3)$ to the hyperbola $5x^2-6y^2-30=0$,then $\tan \theta=$