Circles practice · Medium

Suppose a circle passes through $\left( {2,2} \right)$ and $\left( {9,9} \right)$ and touches the $x$-axis at P. If $O$ is the origin, then $OP$ is equal to

The distinct points $A\left( {0,\;0} \right),\;B\left( {0,\;1} \right),\;C\left( {1,\;0} \right)$ and $D\left( {2a,\;3a} \right)$ are concyclic, then

A circle has two of its diameters along the lines $2 x+$ $3 y-18=0$ and $3 x-y-5=0$ and touches the line $x+2 y+4=0$. Equation of the circle is

Find the equation of the chord of contact of the tangents drawn from $(1,2)$ to the circle $x^2+y^2-2 x+4 y+7=0$.

Find the equation of the chord of the circle $x^2+y^2+6 x+8 y-$ $11=0$, whose middle point is $(1,-1)$.

Three distinct points A, B and C are given in the two-dimensional coordinate plane
such that the ratio of the distance of any one of them from (2, –1) to its distance from
(–1, 5) is 1 : 2. Then the centre of the circle passing through A, B and C is

The number of integral points on the circle, touching the parabola $y^2 = 8x$ at (2, 4) and passing through (0, 4), are equal to

If $f(x+y)=f(x) \cdot f(y)$ for all $x$ and $y, f(1)=2$ and $\alpha_n=f(n), n \in N$, then the equation of the circle having $\left(\alpha_1, \alpha_2\right)$ and $\left(\alpha_3, \alpha_4\right)$ as the ends of its one diameter is

If ${O}$ is the origin and $O P, O Q$ are distinct tangents to the circle $x^2+y^2+2 g x+2 f y+c=0$, the circumcenter of the triangle $O P Q$ is