Parameter from distance condition

7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).

7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.

1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).

6 Points \(A , B\) and \(C\) have coordinates \(A ( - 3,7 ) , B ( 5,1 )\) and \(C ( - 1 , k )\), where \(k\) is a constant.

1. Given that \(A B = B C\), calculate the possible values of \(k\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at \(D\).
2. Calculate the coordinates of \(D\).

9 The points \(A , B\) and \(C\) have coordinates \(( 5,1 ) , ( p , 7 )\) and \(( 8,2 )\) respectively.

1. Given that the distance between points \(A\) and \(B\) is twice the distance between points \(A\) and \(C\), calculate the possible values of \(p\).
2. Given also that the line passing through \(A\) and \(B\) has equation \(y = 3 x - 14\), find the coordinates of the mid-point of \(A B\).

8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).

3 The line \(A B\) has equation \(3 x + 2 y = 7\). The point \(C\) has coordinates \(( 2 , - 7 )\).

1. 1. Find the gradient of \(A B\).
   2. The line which passes through \(C\) and which is parallel to \(A B\) crosses the \(y\)-axis at the point \(D\). Find the \(y\)-coordinate of \(D\).
2. The line with equation \(y = 1 - 4 x\) intersects the line \(A B\) at the point \(A\). Find the coordinates of \(A\).
3. The point \(E\) has coordinates \(( 5 , k )\). Given that \(C E\) has length 5 , find the two possible values of the constant \(k\).

1 The point \(A\) has coordinates \(( - 1,2 )\) and the point \(B\) has coordinates \(( 3 , - 5 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The midpoint of \(A B\) is \(M\).
   1. Find the coordinates of \(M\).
   2. Find an equation of the line which passes through \(M\) and which is perpendicular to \(A B\). [3 marks]
3. The point \(C\) has coordinates \(( k , 2 k + 3 )\). Given that the distance from \(A\) to \(C\) is \(\sqrt { 13 }\), find the two possible values of the constant \(k\).
   [0pt] [4 marks]

2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).

\(A\) is the point \((a, 2a - 1)\) and \(B\) is the point \((2a + 4, 3a + 9)\), where \(a\) is a constant.

1. Find, in terms of \(a\), the gradient of a line perpendicular to \(AB\). [3]
2. Given that the distance \(AB\) is \(\sqrt{260}\), find the possible values of \(a\). [4]