The diameter of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$ is a chord of another circle with center $(2, 1)$. Find the radius of this circle.

The centre$(s)$ of the circle$(s)$ passing through the points $(0, 0)$ and $(1, 0)$ and touching the circle $x^2 + y^2 = 9$ is/are:

If $\theta$ is the acute angle of intersection at a real point of intersection of the circle $x^2 + y^2 = 5$ and the parabola $y^2 = 4x$,then $\tan \theta$ is equal to

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36} + \frac{y^2}{4} = 1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x = 2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$,then $\alpha^2 - \beta^2$ is equal to

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius $5$. If the values of $\lambda$,for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$,are $\lambda_1$ and $\lambda_2$ (where $\lambda_1 < \lambda_2$),then $12\lambda_1 + 29\lambda_2$ is equal to . . . . . . .

Let $r$ be the radius of the circle,which touches the $x$-axis at point $(a, 0)$,where $a < 0$,and the parabola $y^2 = 9x$ at the point $(4, 6)$. Then $r$ is equal to . . . . . . .