The number of points of intersection of $|z - (4 + 3i)| = 2$ and $|z| + |z - 4| = 6$,$z \in \mathbb{C}$ is

The equation of the common tangent to the curves ${y^2} = 8x$ and $xy = -1$ is

Find the length of the common chord of the ellipse $\frac{(x - 2)^2}{9} + \frac{(y + 2)^2}{4} = 1$ and the circle $x^2 + y^2 - 4x + 2y + 4 = 0$.

$A$ circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse,such that the two curves intersect in $4$ points. Let $P$ be any one of their points of intersection. If the major axis of the ellipse is $17$ and the area of the triangle $PF_1F_2$ is $30$,then the distance between the foci is:

For the hyperbola $H : x^{2} - y^{2} = 1$ and the ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ where $a > b > 0$,let $(1)$ the eccentricity of $E$ be the reciprocal of the eccentricity of $H$,and $(2)$ the line $y = \sqrt{\frac{5}{2}} x + K$ be a common tangent of $E$ and $H$. Then $4(a^{2} + b^{2})$ is equal to:

The quadratic equation whose roots are $l$ and $m$,where $l = \lim_{\theta \rightarrow 0} \left( \frac{3 \sin \theta - 4 \sin^2 \theta}{\theta} \right)$ and $m = \lim_{\theta \rightarrow 0} \frac{2 \tan \theta}{\theta(1 - \tan^2 \theta)}$,is: