If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ coincide,then $b^2$ is equal to

If the tangent to the parabola $y^2 = x$ at a point $(\alpha, \beta)$,$(\beta > 0)$ is also a tangent to the ellipse $x^2 + 2y^2 = 1$,then $\alpha$ is equal to

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

Find the common tangent to the curves $x^2 + y^2 = 4$ and $2x^2 + y^2 = 2$.

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$,respectively,and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$,respectively. Then which of the following statements is(are) true?
$(A)$ The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $35$ square units.
$(B)$ The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units.
$(C)$ The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$.
$(D)$ The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$.

The equations of the common tangents to the ellipse $x^2 + 4y^2 = 8$ and the parabola $y^2 = 4x$ are