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?

- [JEE MAIN 2020]

- A
$3 x-4 y-24=0$

- B
$3 x+4 y-6=0$

- C
$4 x+3 y-8=0$

- D
$4 x-3 y+17=0$

The gradient of the tangent line at the point $(a\cos \alpha ,a\sin \alpha )$ to the circle ${x^2} + {y^2} = {a^2}$, is

Tangents drawn from the point $P(1,8)$ to the circle $x^2+y^2-6 x-4 y-11=0$ touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is

- [IIT 2009]

Tangents to a circle at points $P$ and $Q$ on the circle intersect at a point $R$. If $P Q=6$ and $P R=5$, then the radius of the circle is

- [KVPY 2013]

Tangents are drawn from $(4, 4) $ to the circle $x^2 + y^2 - 2x - 2y - 7 = 0$ to meet the circle at $A$ and $B$. The length of the chord $AB $ is

The line $ax + by + c = 0$ is a normal to the circle ${x^2} + {y^2} = {r^2}$. The portion of the line $ax + by + c = 0$ intercepted by this circle is of length