The line $x=8$ is the directrix of the ellipse $\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with the corresponding focus $(2,0)$. If the tangent to $\mathrm{E}$ at the point $\mathrm{P}$ in the first quadrant passes through the point $(0,4\sqrt3)$ and intersects the $x$-axis at $\mathrm{Q}$, then $(3\mathrm{PQ})^{2}$ is equal to ____________.

If the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis and the line…

If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :

If two tangents drawn from a point P to the parabola y2 = 16(x $-$ 3) are at right angles, then the locus of point P is :

Let the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$…

If $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$, ${m_1} \ne {m_2}$ are two common tangents of circle ${x^2} + {y^2} = 2$ and parabola y2 =…