If the eccentricity of the standard hyperbola passing through the point $(4, 6)$ is $2$,then the equation of the tangent to the hyperbola at $(4, 6)$ is

Let $R$ be a rectangle given by the lines $x=0, x=2, y=0$ and $y=5$. Let $A(\alpha, 0)$ and $B(0, \beta)$,where $\alpha \in [0, 2]$ and $\beta \in [0, 5]$,be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4:1$. Then,the mid-point of $AB$ lies on a $.........$.

The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is

The equations of the common tangents to the two hyperbolas $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ are-

If a circle of radius $4 \text{ cm}$ passes through the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{4} = 1$ and is concentric with the hyperbola,then the eccentricity of the conjugate hyperbola of that hyperbola is

$A$ hyperbola passes through the points $(3, 2)$ and $(-17, 12)$ and has its centre at the origin and its transverse axis along the $x$-axis. The length of its transverse axis is: