The equation of three circles are ${x^2} + {y^2} – 12x – 16y + 64 = 0,$ $3{x^2} + 3{y^2} – 36x + 81 = 0$ and ${x^2} + {y^2} – 16x + 81 = 0.$ The co-ordinates of the point from which the length of tangent drawn to each of the three circle is equal is

A circle touches the $y$ -axis at the point $(0,4)$ and passes through the point $(2,0) .$ Which of the following lines is not a tangent to this circle?

If the tangent at a point $P(x,y)$ of a curve is perpendicular to the line that joins origin with the point $P$, then the curve is

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is. . . . . .

The equation of the normal to the circle ${x^2} + {y^2} – 2x = 0$ parallel to the line $x + 2y = 3$ is