If two lines represented by $x^2 - (1 + \sqrt{3})xy + \sqrt{3}y^2 = 0$ make angles $\alpha$ and $\beta$ with the X-axis, then $\tan(\alpha + \beta)$ is...

Let PA and PB be the tangent segments drawn from point P$(6, 8)$ to the circle with the centre at origin O. The radius of circle for which the area of quadrilateral PAOB is maximum, is...

A circle passes through the point $(0,1)$ and touches the parabola $y = x^2$ at the point $(1,1)$. The centre of the circle is...

The eccentricity of the ellipse represented by the equation $7x^2 + 16y^2 - 14x + 64y - 377 = 0$ is...