The length of the chord of the ellipse $\frac{x^2}{4}+y^2=1$ formed on the line $y=x+1$ is

Let $P, Q, R, S$ be the points of intersection of the circle $x^2+y^2=4$ and the hyperbola $x y=\sqrt{3}$. If $P=(\alpha, \beta)$ and $\alpha>\beta>0$, then the equation of the tangent drawn at $P$ to the hyperbola is

The number of values of ' $k$ ' for which the points $(-4,9, k),(-1,6, k),(0,7,10)$ from right-angled isosceles triangle is

A line makes angles $60^{\circ}, 45^{\circ}, \theta$ with positive $X, Y, Z$ axes respectively. If $\theta$ is an acute angle, then $\tan \theta=$