The point of contact of the line $y = x - 1$ with $3{x^2} - 4{y^2} = 12$ is

If the centre, vertex and focus of a hyperbola be $(0, 0), (4, 0)$ and $(6, 0)$ respectively, then the equation of the hyperbola is

The distance between the directrices of a rectangular hyperbola is $10$ units, then distance between its foci is

Area of the triangle formed by the lines $x -y = 0, x + y = 0$ and any tangent to the hyperbola $x^2 -y^2 = a^2$ is :-

The normal to the rectangular hyperbola $xy = c^2$ at the point $'t_1'$ meets the curve again at the point $'t_2'$ . Then the value of $t_{1}^{3} t_{2}$ is

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to