Midpoint of line segment

5 The equation of a curve is \(y = x ^ { 2 } - 4 x + 7\) and the equation of a line is \(y + 3 x = 9\). The curve and the line intersect at the points \(A\) and \(B\).

1. The mid-point of \(A B\) is \(M\). Show that the coordinates of \(M\) are \(\left( \frac { 1 } { 2 } , 7 \frac { 1 } { 2 } \right)\).
2. Find the coordinates of the point \(Q\) on the curve at which the tangent is parallel to the line \(y + 3 x = 9\).
3. Find the distance \(M Q\).

2 Points \(A , B\) and \(C\) have coordinates \(( 2,5 ) , ( 5 , - 1 )\) and \(( 8,6 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find the equation of the line through \(C\) perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\).

4 The line \(4 x + k y = 20\) passes through the points \(A ( 8 , - 4 )\) and \(B ( b , 2 b )\), where \(k\) and \(b\) are constants.

1. Find the values of \(k\) and \(b\).
2. Find the coordinates of the mid-point of \(A B\).

7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.

1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
2. Find the set of values of \(m\) for which \(L\) does not meet the curve.

1. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\).

Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).

2 A is the point \(( 1,5 )\) and B is the point \(( 6 , - 1 )\). M is the midpoint of AB . Determine whether the line with equation \(y = 2 x - 5\) passes through M.

6. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\). Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).

1 The triangle \(A B C\) has vertices \(A ( - 2,3 ) , B ( 4,1 )\) and \(C ( 2 , - 5 )\).

1. Find the coordinates of the mid-point of \(B C\).
2. 1. Find the gradient of \(A B\), in its simplest form.
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(x + q y = r\), where \(q\) and \(r\) are integers.
   3. Find an equation of the line passing through \(C\) which is parallel to \(A B\).
3. Prove that angle \(A B C\) is a right angle.

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

1. Find the coordinates of the mid-point of \(AB\). [2]
2. Find, in the form \(y = mx + c\), an equation for the straight line through \(A\) and \(B\). [4]

A is the point \((1, 5)\) and B is the point \((6, -1)\). M is the midpoint of AB. Determine whether the line with equation \(y = 2x - 5\) passes through M. [3]