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1. Find the gradient of the line that joins the points $(3a, -2a)$ and $(7a, 2a)$.
2. The line joining the points $(7,2)$ and $(b, 3b)$ has gradient $4$. Find $b$.
3. A line with gradient 3 and $y$-intercept $(0,5)$ has the equation $ax-2y+c=0$. Find $a$ and $c$.
4. Find the midpoint of the following pairs of points:
  1. $(4,1)$ and $(6,9)$
  2. $(-2,-7)$ and $(-5,4)$
5. Find the distance between the following pairs of points:
  1. $(2,3)$ and $(-3,2)$
  2. $(1,5)$ and $(-6,-3)$
6. Find the equation of the line perpendicular to the line $y=6x-9$ and passes through the point $(0,1)$
7. The lines $y=4x-10$ and $y=x-1$ intersect. Find the equation of the line that passes through this point of intersection and has a gradient of 2.
8. A line passes through the point $(0,6)$ and has gradient $-2$. It intersects the line with equation $5x-8y-15=0$. Find the point of intersection.
9. Find the equation of the line that passes through the points $(-4,-1)$ and $(6,4)$. Write your answer in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are integers.
10. The line that passes through the points $(0,4)$ and $(-3,0)$ is perpendicular to the line that passes through the points $(-6,0)$ and $(0,c)$. Find $c$.
11. The lines $y=x-5$ and $y=3x-13$ intersect at the point $P$. The point $Q$ has coordinates $(4,2)$. Find the equation of the line that passes through both $P$ and $Q$
12. Find the equation of the line perpendicular to the line $4x-6y+2=0$ and passes through the point $(3,4)$.
13. The line $l$ passes through the points $(-3,0)$ and $(3,-2)$. The line $m$ passes through the points $(1,8)$ and $(-1,2)$. Show that these lines are perpendicular.
14. The distance between the two points $(2,x)$ and $(5,7)$ is $3\sqrt{10}$. Find two possible values of $x$.
15. Find the equation of the perpendicular bisector of the following pairs of points in the form $y = mx+c$:
  1. $(3,5)$ and $(5,9)$
  2. $(-1,-4)$ and $(-7,8)$
16. A triangle goes through the points $P(3,k)$, $Q(6,8)$ and $R(10,10)$. Angle $PQR$ is a right angle.
Find the equation of the line that goes through $P$ and $R$.
17. A straight line passes through the points $(a,4)$ and $(3a,3)$. It has the equation $x+6y+c=0$. Find $a$ and $c$.
18. The vertices of a quadrilateral $ABCD$ have coordinates $A(-1,5)$, $B(7,1)$, $C(5,-3)$, and $D(-3,1)$. Show that this is a rectangle.
19. The line $l_1$ passes through the point $A(2, 5)$ and has gradient $-\frac{1}{2}$.
  1. Find an equation of $l_1$, giving your answer in the form $y = mx + c$.
  2. Show that the point $B(-2, 7)$ lies on $l_1$.
  3. Find the length of $AB$ in the form $k\sqrt{5}$.
  4. The point $C$ lies on $l_1$ and has $x$ coordinate equal to $p$. The length of $AC$ is 5. Show that $p$ satisfies $p^2-4p-16=0$.
20. A point $P$ lies on the line $y=4-3x$. The distance between the point $P$ and the origin is $\sqrt{34}$. Find two possible positions of point $P$.
21. Find the equation of the line that passes through the points $(7,2)$ and $(9,-8)$. Give your answer in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are integers.
22. The line $l_1$ passes through the points $P(-1, 2)$ and $Q(11, 8)$.
  1. Find an equation for $l_1$ in the form $y = mx + c$.
  2. The line $l_2$ passes through the point $R(10, 0)$ and is perpendicular to $l_1$. $l_1$ and $l_2$ intersect at the point $S$. Find the length of $RS$.
  3. Find the area of triangle $PQR$.
23. The points $\left(0, -\frac{2}{3}\right)$, $\left(1, -\frac{1}{3}\right)$ and $\left(1, \frac{1}{3}\right)$ all lie on the circumference of a circle. Find the center of the circle.
24. The points $A$ and $B$ have coordinates $(2, -3)$ and $(8, 5)$ respectively, and $AB$ is a chord of a circle with centre $(4,y)$. Find the radius of the circle.