The locus of the midpoints of the chords of the hyperbola $x^{2}-y^{2}=4$,which touch the parabola $y^{2}=8x$,is:

If the points of intersection of two distinct conics $x^2+y^2=4b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3x^2$,then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is............................

If $e_{1}$ and $e_{2}$ are the eccentricities of the ellipse $\frac{x^{2}}{18}+\frac{y^{2}}{4}=1$ and the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ respectively,and $(e_{1}, e_{2})$ is a point on the ellipse $15x^{2}+3y^{2}=k$,then $k$ is equal to:

$A$ conic passes through the point $(2, 4)$ and is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected at the point of tangency. Then the foci of the conic are:

Let $e_{1}$ and $e_{2}$ be the eccentricities of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1$ $(b < 5)$ and the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{b^{2}}=1$ respectively,satisfying $e_{1}e_{2}=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively,then the ordered pair $(\alpha, \beta)$ is equal to

$AB$ is a chord of the parabola $y^2 = 4ax$ with one endpoint $A$ at the vertex of the parabola. $BC$ is drawn perpendicular to $AB$ and meets the axis of the parabola at $C$. What is the projection of $BC$ on the axis of the parabola?