The equation of the circle symmetric to the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ about the line $x - y = 3$ is

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is

The vertex of the parabola $(y - 1)^2 = 8(x - 1)$ is at the centre of a circle and the parabola cuts that circle at the ends of its latus rectum. Then the equation of that circle is

Consider two curves $C_1 : y^2 = 2x$ and $C_2 : x^2 + y^2 - 3x + 2 = 0$. Then,

Let $P(3 \cos \alpha, 2 \sin \alpha)$,$\alpha \neq 0$,be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$,$Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$,and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $(2 + \cos \alpha, 3 + \frac{2}{3} \sin \alpha)$. Then the sum of the ordinates of all possible points $R$ is:

If the length of the tangents drawn from the point $(1, 2)$ to the circles $x^2 + y^2 + x + y - 4 = 0$ and $3x^2 + 3y^2 - x - y + k = 0$ are in the ratio $4 : 3$,then $k =$