The foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the hyperbola $\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$ coincide. Then the value of $b^2$ is

$P(6, 3)$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ . If the normal at point $P$ intersect the $x-$ axis at $(10, 0)$ , then the eccentricity of the hyperbola is

Tangents are drawn to the hyperbola $4{x^2} - {y^2} = 36$ at the points $P$ and $Q.$ If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of $\Delta PTQ$ is :

The locus of middle points of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is

A common tangent $T$ to the curves $C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C _{1}$ at ( $\left.x _{1}, y _{1}\right)$ and $C _{2}$ at $\left( x _{2}, y _{2}\right)$, then $\left|2 x _{1}+ x _{2}\right|$ is equal to $......$

The vertices of a hyperbola are at $(0, 0)$ and $(10, 0)$ and one of its foci is at $(18, 0)$. The equation of the hyperbola is