Coordinate geometry of the circle

Solutions to this question by accurate drawing will not be accepted.

A circle $C_1$ has centre $(3, 5)$.
The point $P(1, 4)$ lies on circle $C_1$.

A circle $C_2$ has centre $(12, 8)$.
The line with equation $y = 19 - 3x$ is the common chord of circles $C_1$ and $C_2$.

Find the radius of circle $C_2$.

A circle has equation $x^2 + y^2 - 25 = 0$.
A second circle has the same radius as the first circle, and the coordinates of its centre are both positive.
The two circles intersect at the points $A$ and $B$.
The line $AB$ has length 6 and is parallel to the line $y = -x$.

Find the equation of the second circle in the form $x^2 + y^2 + ax + by + c = 0$, where $a$, $b$ and $c$ are constants.

The line with equation $x + 3y = k$, where $k$ is a positive constant, is a tangent to the curve with equation $x^2 + y^2 + 2y - 9 = 0$. Find the value of $k$ and hence find the coordinates of the point where the line touches the curve.

Find the exact values of the constant $k$ for which the line $y = 2x + 1$ is a tangent to the curve

The line $y=x-5$ meets the curve $x^2+y^2+2x-35=0$ at the points $A$ and $B$. Find the exact length of $AB$.

Solutions to this question by accurate drawing will not be accepted.

The diagram shows the quadrilateral $ABCD$ in which $A$ is the point $(4, 2)$ and $B$ is the point $(-2, -10)$. The points $C$ and $D$ lie on the line $x = 14$. The diagonal $AC$ is perpendicular to $AB$ and passes through the mid-point, $M$, of the diagonal $BD$. Find the area of the quadrilateral $ABCD$.

Solutions to this question by accurate drawing will not be accepted.

In the diagram the points $A(-1, 5)$, $B(-2, 6)$, $C(4, 10)$ and $D$ are the vertices of a quadrilateral in which $AD$ is parallel to the $x$-axis. The perpendicular bisector of $BC$ passes through $D$. Find the area of the quadrilateral $ABCD$.

The length of $BD$ is $\sqrt{125}$. Find the coordinates of the two possible positions of point $D$.

Find the equation of the line $L$, the perpendicular bisector of the line $AB$.