If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$

The number of real values of $\alpha$ for which the pair of lines represented by $\left(\alpha^2+12|\alpha|\right) x^2+6 x y+(18-21|\alpha|) y^2=0$ are at right angles to each other, is

The line $x+2 y-c=0$ meets the curve $x^2+y^2-3 x-6 y+3=0$ at two points $P$ and $Q$ and $\angle P O Q=\frac{\pi}{2}$, where $O$ is the origin. Then, $2 c^2-15 c=$

Let $A B C$ be a triangle. Let a point $P$ divide $A B$ in the ratio $1: 2$ internally and a point $Q$ divide $B C$ in the ratio $1: 2$ internally. Let $D$ be the point of intersection of $A Q$ and $C P$. If the area of the $\triangle A B C$ is $k$ square units, then the area of the $\triangle B C D$ in (sq. units) is