Let $\alpha, \beta$ be the roots of the quadratic equation $x^2 + px + p^3 = 0$ $(p \neq 0)$. If $(\alpha, \beta)$ is a point on the parabola $y^2 = x$,then the roots of the quadratic equation are:

If the foci of $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ and $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ coincide,then the value of $a$ is

An ellipse intersects the hyperbola $2x^2 - 2y^2 = 1$ orthogonally. The eccentricity of the ellipse is the reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes,then:
$(A)$ Equation of ellipse is $x^2 + 2y^2 = 2$
$(B)$ The foci of ellipse are $(\pm 1, 0)$
$(C)$ Equation of ellipse is $x^2 + 2y^2 = 4$
$(D)$ The foci of ellipse are $(\pm \sqrt{2}, 0)$

If the curves $\frac{x^2}{\alpha} + \frac{y^2}{4} = 1$ and $y^3 = 16x$ intersect at right angles,then a value of $\alpha$ is

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 $(-6,0)$.

Let a line $L: 2x + y = k, k > 0$ be a tangent to the hyperbola $x^2 - y^2 = 3$. If $L$ is also a tangent to the parabola $y^2 = \alpha x$,then $\alpha$ is equal to: