Incident ray is along the unit vector $$\hat{v}$$ and the reflected ray is along the unit vector $$\widehat{w}$$. The normal is along unit vector $$\hat{a}$$ outwards. Express $$\hat{w}$$, in terms of $$\hat{a}$$ and $$\hat{v}$$.

Tangents are drawn from any point on the hyperbola $$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$ to the circle $$x^{2}+y^{2}=9$$. Find the locus of mid-point of the chord of contact.

Find the equation of the common tangent in the first quadrant to the circle $$x^{2}+y^{2}=16$$ and the ellipse $$\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$$. Also find the length of the intercept of the tangent between the coordinate axes.

If length of tangent at any point on the curve $$y = f(x)$$ intercepted between the point and the X-axis is of length 1. Find the equation of the curve.