Points A and B have coordinates (8, 3) and (p, q) respectively. The equation of the perpendicular bisector of AB is y = -2x + 4. Find the values of p and q.

The coordinates of points A and B are \((-3k - 1, k + 3)\) and \((k + 3, 3k + 5)\) respectively, where \(k\) is a constant \((k \neq -1)\).

1. Find and simplify the gradient of \(AB\), showing that it is independent of \(k\).
2. Find and simplify the equation of the perpendicular bisector of \(AB\).

Points A and B have coordinates \((h, h)\) and \((4h + 6, 5h)\) respectively. The equation of the perpendicular bisector of \(AB\) is \(3x + 2y = k\). Find the values of the constants \(h\) and \(k\).

A curve is defined by the equation \(y = \frac{1}{x} + c\) and a line is defined by the equation \(y = cx - 3\), where \(c\) is a constant.

(i) Determine the set of values of \(c\) for which the curve and the line intersect.

(ii) The line is tangent to the curve for two specific values of \(c\). For each of these values, find the \(x\)-coordinate of the point where the tangent touches the curve.

The points A (1, 1) and B (5, 9) lie on the curve \(6y = 5x^2 - 18x + 19\).

(i) Show that the equation of the perpendicular bisector of AB is 2y = 13 - x.

The perpendicular bisector of AB meets the curve at C and D.

(ii) Find, by calculation, the distance CD, giving your answer in the form \(\sqrt{\frac{p}{q}}\), where p and q are integers.

Given two points, \(A(-1, 1)\) and \(P(a, b)\), where \(a\) and \(b\) are constants, the gradient of \(AP\) is 2.

1. Find an expression for \(b\) in terms of \(a\).
2. Point \(B(10, -1)\) is such that \(AP = AB\). Calculate the coordinates of the possible positions of \(P\).

The point A has coordinates (-2, 6). The equation of the perpendicular bisector of the line AB is given by:

\(2y = 3x + 5\).

(i) Find the equation of line AB.

(ii) Find the coordinates of point B.

Three points, A, B, and C, are such that B is the midpoint of AC. The coordinates of A are (2, m) and the coordinates of B are (n, -6), where m and n are constants.

1. Find the coordinates of C in terms of m and n.
2. The line y = x + 1 passes through C and is perpendicular to AB. Find the values of m and n.

The line \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) and \(b\) are positive constants, intersects the x- and y-axes at the points \(A\) and \(B\) respectively. The mid-point of \(AB\) lies on the line \(2x + y = 10\) and the distance \(AB = 10\). Find the values of \(a\) and \(b\).

C is the midpoint of the line segment joining A(14, -7) and B(-6, 3). The line through C is perpendicular to AB and crosses the y-axis at D.

(i) Find the equation of the line CD in the form y = mx + c.

(ii) Find the distance AD.

Triangle ABC has vertices at A (-2, -1), B (4, 6), and C (6, -3).

1. Show that triangle ABC is isosceles and find the exact area of this triangle.
2. The point D is on AB such that CD is perpendicular to AB. Calculate the x-coordinate of D.

The equation of a curve is \(y = (x - 3)\sqrt{x + 1} + 3\). The following points lie on the curve. Non-exact values are rounded to 4 decimal places.

\(A (2, k)\) \(B (2.9, 2.8025)\) \(C (2.99, 2.9800)\) \(D (2.999, 2.9980)\) \(E (3, 3)\)

1. Find \(k\), giving your answer correct to 4 decimal places.
2. Find the gradient of \(AE\), giving your answer correct to 4 decimal places.

The gradients of \(BE, CE\) and \(DE\), rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.

3. State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point \(E\).

Three points have coordinates \(A(0, 7)\), \(B(8, 3)\), and \(C(3k, k)\). Find the value of the constant \(k\) for which:

1. \(C\) lies on the line that passes through \(A\) and \(B\).
2. \(C\) lies on the perpendicular bisector of \(AB\).

Two points have coordinates \(A(5, 7)\) and \(B(9, -1)\).

(i) Find the equation of the perpendicular bisector of \(AB\).

The line through \(C(1, 2)\) parallel to \(AB\) meets the perpendicular bisector of \(AB\) at the point \(X\).

(ii) Find, by calculation, the distance \(BX\).

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

(i) Given that AB = BC, calculate the possible values of k.

The perpendicular bisector of AB intersects the x-axis at D.

(ii) Calculate the coordinates of D.

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

(i) If the distance \(AB\) is 13 units, find the possible values of \(p\).

(ii) If the line with equation \(2x + 3y = 9\) is perpendicular to \(AB\), find the value of \(p\).

The point C lies on the perpendicular bisector of the line joining the points A (4, 6) and B (10, 2). C also lies on the line parallel to AB through (3, 11).

1. Find the equation of the perpendicular bisector of AB.
2. Calculate the coordinates of C.

The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Find the area of triangle \(AOB\) in terms of \(t\).
2. The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\). Show that the midpoint of \(PC\) lies on the line \(y = x\).

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

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

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

1. Find the values of k and b.
2. Find the coordinates of the mid-point of AB.

Determine the coordinates where the perpendicular bisector of the line segment connecting the points (2, 7) and (10, 3) intersects the x-axis.

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

Point M is the midpoint of the line segment joining the points (3, 7) and (-1, 1). Find the equation of the line passing through M that is parallel to the line \(\frac{x}{3} + \frac{y}{2} = 1\).

The point A has coordinates (3, 1) and the point B has coordinates (-21, 11). The point C is the midpoint of AB.

1. Find the equation of the line through A that is perpendicular to y = 2x - 7.
2. Find the distance AC.

The point A has coordinates (-1, 6) and the point B has coordinates (7, 2).

(i) Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx + c.

(ii) A point C on the perpendicular bisector has coordinates (p, q). The distance OC is 2 units, where O is the origin. Write down two equations involving p and q and hence find the coordinates of the possible positions of C.

Find the coordinates of the reflection of the point (-1, 3) across the line 3y + 2x = 33.

A curve is given by the equation \(y = x^2 - 4x + 4\) and a line by the equation \(y = mx\), where \(m\) is a constant. For \(m = 1\), the curve and the line intersect at points \(A\) and \(B\). Find the coordinates of the midpoint of \(AB\).

The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\). In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\).

The point A has coordinates (-1, -5) and the point B has coordinates (7, 1). The perpendicular bisector of AB meets the x-axis at C and the y-axis at D. Calculate the length of CD.

The coordinates of point A are (-3, 2) and the coordinates of point C are (5, 6). The midpoint of AC is M, and the perpendicular bisector of AC intersects the x-axis at B.

1. Find the equation of MB and the coordinates of B.
2. Show that AB is perpendicular to BC.
3. Given that ABCD is a square, find the coordinates of D and the length of AD.

A line has the equation \(y = kx + 6\) and a curve has the equation \(y = x^2 + 3x + 2k\), where \(k\) is a constant. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(AB\) and the coordinates of the midpoint of \(AB\).

The line \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) and \(b\) are positive constants, meets the x-axis at \(P\) and the y-axis at \(Q\). Given that \(PQ = \sqrt{45}\) and that the gradient of the line \(PQ\) is \(-\frac{1}{2}\), find the values of \(a\) and \(b\).

The line \(L_1\) passes through the points \(A(2, 5)\) and \(B(10, 9)\). The line \(L_2\) is parallel to \(L_1\) and passes through the origin. The point \(C\) lies on \(L_2\) such that \(AC\) is perpendicular to \(L_2\). Find:

1. the coordinates of \(C\),
2. the distance \(AC\).

The coordinates of two points A and B are (1, 3) and (9, -1) respectively, and D is the midpoint of AB. A point C has coordinates (x, y), where x and y are variables.

1. State the coordinates of D.
2. Given that CD² = 20, write an equation relating x and y.
3. Given that AC and BC are equal in length, find an equation relating x and y and show that it can be simplified to y = 2x - 9.
4. Using the results from parts (ii) and (iii), find the possible coordinates of C.

The line \(x - y + 4 = 0\) intersects the curve \(y = 2x^2 - 4x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \((3, 7)\).

(ii) Find the coordinates of \(Q\).

(iii) Find the equation of the line joining \(Q\) to the mid-point of \(AP\).

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

(i) Find the coordinates of the midpoint of AB.

(ii) Find the equation of the line through C perpendicular to AB. Give your answer in the form ax + by + c = 0.

The curve \(y^2 = 12x\) intersects the line \(3y = 4x + 6\) at two points. Find the distance between the two points.

Three points have coordinates \(A(2, 6)\), \(B(8, 10)\), and \(C(6, 0)\). The perpendicular bisector of \(AB\) meets the line \(BC\) at \(D\). Find:

1. the equation of the perpendicular bisector of \(AB\) in the form \(ax + by = c\),
2. the coordinates of \(D\).

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

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

The curve \(y = 9 - \frac{6}{x}\) and the line \(y + x = 8\) intersect at two points. Find:

1. the coordinates of the two points,
2. the equation of the perpendicular bisector of the line joining the two points.

The line \(L_1\) has the equation \(2x + y = 8\). The line \(L_2\) passes through the point \(A(7, 4)\) and is perpendicular to \(L_1\).

1. Find the equation of \(L_2\).
2. Given that the lines \(L_1\) and \(L_2\) intersect at the point \(B\), find the length of \(AB\).

The curve \(C_1\) has the equation \(y = x^2 - 4x + 7\). The curve \(C_2\) has the equation \(y^2 = 4x + k\), where \(k\) is a constant. The tangent to \(C_1\) at the point where \(x = 3\) is also the tangent to \(C_2\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).

Two points A and B have coordinates (1, 3) and (9, -1) respectively. The perpendicular bisector of AB intersects the y-axis at the point C. Find the coordinates of C.

Two points A and B have coordinates (-1, 1) and (3, 4) respectively. The line BC is perpendicular to AB and intersects the x-axis at C.

1. Find the equation of BC and the x-coordinate of C.
2. Find the distance AC, giving your answer correct to 3 decimal places.

The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.

In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).

(ii) Find the coordinates of \(A\) and \(B\).

(iii) Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\).

Two points A and B have coordinates \((3a, -a)\) and \((-a, 2a)\) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(AB\).
2. The length of the line \(AB\) is \(3\frac{1}{3}\) units. Find the value of \(a\).