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 = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

The length of the line segment $P Q$ 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 $M = (\frac{1 + 4}{2}, \frac{2 + 6}{2}) = (2.5, 4)$
Length $A B = \sqrt{(4 - 1)^{2} + (6 - 2)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$

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 $C D = \frac{7 - 3}{4 - 2} = 2$
Gradient of $E F = \frac{6 - 2}{3 - 1} = 2$
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: $y = 3 x - 6 + 1 = 3 x - 5$

Quiz 1

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Quiz 2

Question 66

A curve has equation $y = x^{2} + 2 c x + 4$ and a straight line has equation $y = 4 x + c$ , 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 $y = 3 x - 20$ 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 $y = 3 x - 20$ is a tangent to the circle.

Question 68

A line has equation $y = 3 x + k$ and a curve has equation $y = x^{2} + k x + 6$ , 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 $x^{2} + y^{2} - 4 x + 6 y - 77 = 0$ .

(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 $A B$ is $y = - 2 x + 4$ . 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 $y = m x - 6$ is a tangent to the curve with equation $y = x^{2} - 4 x + 3$ . 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 $A B C = 90^{\circ}$ .
(b) Hence state the coordinates of $D$ .
(c) Find an equation of the circle.

Question 75

A line has equation $y = m x + c$ and a curve has equation $y = x^{2} + k x + 6$ , 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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