Let $$y=y(x)$$ be the solution curve of the differential equation $$ \frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}$$, $$x >1$$ passing through the point $$\left(2, \sqrt{\frac{1}{3}}\right)$$. Then $$\sqrt{7}\, y(8)$$ is equal to :

The differential equation of the family of circles passing through the points $$(0,2)$$ and $$(0,-2)$$ is :

Let the tangents at two points $$\mathrm{A}$$ and $$\mathrm{B}$$ on the circle $$x^{2}+\mathrm{y}^{2}-4 x+3=0$$ meet at origin $$\mathrm{O}(0,0)$$. Then the area of the triangle $$\mathrm{OAB}$$ is :

Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. A parabola is drawn whose focus is same as the focus of $$\mathrm{H}$$ with positive abscissa and the directrix of the parabola passes through the other focus of $$\mathrm{H}$$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $$\mathrm{H}$$, where e is the eccentricity of H, then which of the following points lies on the parabola?