If the radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangle L M N$ is

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse $9 x^2+4 y^2=72$ at the point $(2,3)$ with the $X$-axis is

If $3 \sqrt{2} x-4 y=12$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{5}{4}$ is its eccentricity, then $a^2-b^2=$