The equation of the hyperbola whose foci are $(6, 4)$ and $(-4, 4)$ and eccentricity $2$ is given by
$12{x^2} - 4{y^2} - 24x + 32y - 127 = 0$
$12{x^2} + 4{y^2} + 24x - 32y - 127 = 0$
$12{x^2} - 4{y^2} - 24x - 32y + 127 = 0$
$12{x^2} - 4{y^2} + 24x + 32y + 127 = 0$
The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. The point of the tangency being $(a, b)$ then find the value of $\sin ^{-1}\left(\frac{a}{b}\right)$
For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remains constant with change in $'\alpha '$
Curve $xy = {c^2}$ is said to be
The eccentricity of a hyperbola passing through the points $(3, 0)$, $(3\sqrt 2 ,\;2)$ will be
Let the equation of two diameters of a circle $x ^{2}+ y ^{2}$ $-2 x +2 fy +1=0$ be $2 px - y =1$ and $2 x + py =4 p$. Then the slope $m \in(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to $......$