The equation of normal to the circle $2{x^2} + 2{y^2} - 2x - 5y + 3 = 0$ at $(1, 1)$ is

Let $O$ be the centre of the circle $x ^2+ y ^2= r ^2$, where $r >\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x+2 y=4$, then the value of $r$ is. . . .

The equations of the tangents to circle $5{x^2} + 5{y^2} = 1$, parallel to line $3x + 4y = 1$ are

Length of the tangent drawn from any point on the circle ${x^2} + {y^2} + 2gx + 2fy + {c_1} = 0$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is

If the straight line $y = mx + c$ touches the circle ${x^2} + {y^2} - 2x - 4y + 3 = 0$ at the point $(2, 3)$, then $c =$

The line $2x - y + 1 = 0$ is tangent to the circle at the point $(2, 5)$ and the centre of the circles lies on $x-2y=4$. The radius of the circle is