If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2}\, = \,{k^2}$$ orthogonally, then the equation of the locus of its centre is

The equations of the tangents drawn from the origin to the circle $${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$$, are

If $${y^2} = P\left( x \right)$$, a polynomial of degree $$3$$, then $$2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)$$ equals

If the angles of a triangle are $${30^ \circ }$$ and $${45^ \circ }$$ and the included side is $$\left( {\sqrt 3 + 1} \right)$$ cms, then the area of the triangle is ...............