If $$y$$ is a function of $$x$$ and log $$(x+y)-2xy=0$$, then the value of $$y'(0)$$ is equal to

If the line $$62x + \sqrt 6 y = 2$$ touches the hyperbola $${x^2} - 2{y^2} = 4$$, then the point of contact is

If tangents are drawn to the ellipse $${x^2} + 2{y^2} = 2,$$ then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

The angle between the tangents drawn from the point $$(1, 4)$$ to the parabola $${y^2} = 4x$$ is