If $X$ is a Poisson variate such that $\frac{5}{3} k=P(X=2) =P(X=3)$, then $P(X=5)=$

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on $Y$ - axis such that $C D=4$ and $C$ is a point below $D$. Then, the locus of the point of intersection of the lines $A C$ and $B D$ is

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

In an isosceles triangle the ends of its base are $(2 a, 0),(0, a)$ and one of its two other sides is a horizontal line other than $X$-axis. If the third vertex is $\left(x_1, y_1\right)$, then $x_1+y_1=$