Coordinate Geometry AS & A level

Coordinate Geometry

1. Midpoint and Length of a Line Segment

The midpoint \(M\) of a line segment joining points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by:

\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)

The length of the line segment \(PQ\) is:

\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

2. Gradient of a Line

The gradient (slope) of the line joining the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is:

\( \frac{y_2 - y_1}{x_2 - x_1} \)

3. Parallel and Perpendicular Lines

For parallel lines, the gradients \(m_1\) and \(m_2\) are equal:

\( m_1 = m_2 \)

For perpendicular lines, the product of their gradients is:

\( m_1 \times m_2 = -1 \)

4. Equation of a Straight Line

The equation of a straight line with gradient \(m\) passing through point \((x_1, y_1)\) is:

\( y - y_1 = m(x - x_1) \)

Example Problems and Solutions

Example 1

Problem: Find the midpoint and length of the line segment joining points \(A(1, 2)\) and \(B(4, 6)\).

Solution:

• Midpoint \(M = \left(\frac{1 + 4}{2}, \frac{2 + 6}{2}\right) = (2.5, 4)\)
• Length \(AB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Example 2

Problem: Determine if the lines through points \(C(2, 3)\) and \(D(4, 7)\), and points \(E(1, 2)\) and \(F(3, 6)\) are parallel.

Solution:

• Gradient of \(CD = \frac{7 - 3}{4 - 2} = 2\)
• Gradient of \(EF = \frac{6 - 2}{3 - 1} = 2\)
• Since both gradients are equal, the lines are parallel.

Example 3

Problem: Find the equation of a line with gradient 3 passing through the point \(G(2, 1)\).

Solution:

• Using the formula \(y - y_1 = m(x - x_1)\):
• \(y - 1 = 3(x - 2)\)
• Equation: \(y = 3x - 6 + 1 = 3x - 5\)

Quiz 1

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Quiz 2

Question 66

A curve has equation \(y = x^2 + 2cx + 4\) and a straight line has equation \(y = 4x + c\), where \(c\) is a constant. Find the set of values of \(c\) for which the curve and line intersect at two distinct points.

Question 67

The circle with equation \((x+1)^2 + (y-2)^2 = 85\) and the straight line with equation \(y = 3x - 20\) are shown in the diagram. The line intersects the circle at points \(A\) and \(B\), and the center of the circle is at point \(C\).

 
• (a) Find, by calculation, the coordinates of \(A\) and \(B\).
 
• (b) Find an equation of the circle which has its center at \(C\) and for which the line with equation \(y = 3x - 20\) is a tangent to the circle.

Question 68

A line has equation \(y = 3x + k\) and a curve has equation \(y = x^2 + kx + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.

Question 69

The points \(A(7,1)\), \(B(7,9)\), and \(C(1,9)\) are on the circumference of a circle.

 
• (a) Find an equation of the circle.
 
• (b) Find an equation of the tangent to the circle at \(B\).

Question 70

The equation of a circle is \(x^2 + y^2 - 4x + 6y - 77 = 0\).

 
• (a) Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
 
• (b) Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

Question 71

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 \(q\).

Question 72

The point \(A\) has coordinates \((1,5)\) and the line \(l\) has gradient \(-\frac{2}{3}\) and passes through \(A\). A circle has center \((5,11)\) and radius \(\sqrt{52}\).

 
• (a) Show that \(l\) is the tangent to the circle at \(A\).
 
• (b) Find the equation of the other circle of radius \(\sqrt{52}\) for which \(l\) is also the tangent at \(A\).

Question 73

A line with equation \(y = mx - 6\) is a tangent to the curve with equation \(y = x^2 - 4x + 3\). Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.

Question 74

Points \(A(-2,3)\), \(B(3,0)\), and \(C(6,5)\) lie on the circumference of a circle with center \(D\).

 
• (a) Show that angle \(ABC = 90^\circ\).
 
• (b) Hence state the coordinates of \(D\).
 
• (c) Find an equation of the circle.

Question 75

A line has equation \(y = mx + c\) and a curve has equation \(y = x^2 + kx + 6\), where \(m\), \(c\), and \(k\) are constants. Find the set of values of \(m\) and \(c\) for which the line is a tangent to the curve.

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