If the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$ are the same as the foci of the hyperbola $\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}$,then $b^2 = \dots$

If the tangent drawn to the parabola $y^2=4x$ at $(t^2, 2t)$ is the normal to the ellipse $4x^2+5y^2=20$ at $(\sqrt{5} \cos \theta, 2 \sin \theta)$,then

The slopes of the common tangents to the parabola $(x - 1)^2 = 4(y - 2)$ and the ellipse $\frac{(x - 1)^2}{1} + \frac{(y - 2)^2}{2} = 1$ are $m_1$ and $m_2$. Then,$m_1^2 + m_2^2$ is equal to:

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:

The chord $PQ$ of the rectangular hyperbola $xy = a^2$ meets the $x$-axis at $A$. If $C(h, k)$ is the midpoint of $PQ$ and $O$ is the origin,then the $\Delta ACO$ is:

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