If the common tangents to the parabola, x² = 4y and the circle, x² + y² = 4 intersect at the point P, then the distance of P from the origin, is :

P and Q are two distinct points on the parabola, y² = 4x, with parameters t and t₁ respectively. If the normal at P passes through Q, then the minimum value of $$t_1^2$$ is :

Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the circle, $${x^2} + {\left( {y + 6} \right)^2} = 1$$. Then the equation of the circle, passing through $$C$$ and having its centre at $$P$$ is:

Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio $$1:3$$, then locus of $$P$$ is :