The equation of the circle passing through the origin and cutting the circles $x^{2}+y^{2}+6 x-15=0$ and $x^{2}+y^{2}-8 y-10=0$ orthogonally is

$S=(-1,1)$ is the focus, $2 x-3 y+1=0$ is the directrix corresponding, to $S$ and $\frac{1}{2}$ is the eccentricity of an ellipse, If $(a, b)$ is the centre of the ellipse, then $3 a+2 b$ :

$S=y^{2}-4 a x=0, S^{\prime}=y^{2}+a x=0$ are two parabolas and $P(t)$ is a point on the parabola $S^{\prime}=0$. If $A$ and $B$ are the feet of the perpendiculars from $P$ on to coordinate $2 x_{4}$ and $A B$ is a tangent to the parabola $S=0$ at the point $Q\left(t_{1}\right)$, then $t_{1}=$

$a$ and $b$ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes, If its latus rectum is of length 4 units and the distance between its foci is $4 \sqrt{2}$, then $a^{2}+b^{2}=$