Coordinate Geometry Review Flashcards

Summary of coordinate geometry concepts including line segments, gradients, straight lines, and circle equations for A-Level Mathematics.

Midpoint formula

The formula used to find the midpoint $M$ of a line segment joining points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is

$$M=\left(\frac{x1+x2}{2},\frac{y1+y2}{2}\right)$$

Length of a line segment

The formula for the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is

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

Collinear points

Points that lie on the same straight line, such that the gradient between any two of the points is equal.

Parallel lines

Two lines are parallel if their gradients are equal, represented by the rule $m_1 = m_2$.

Perpendicular lines

Two lines are perpendicular if the product of their gradients is $-1$, represented by the rule $m_1\cdot m_2=-1$

Gradient formula

The steepness of a line segment joining points $P(x_1, y_1)$ and $Q(x_2, y_2)$,

calculated as $m=\frac{y2-y1}{x2-x1}$

Equation of a straight line (Point-Gradient Form)

The equation of a straight line with gradient $m$ that passes through point $(x_1, y_1)$ is

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

Equation of a straight line (Gradient-Intercept Form)

The equation $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.

Perpendicular bisector

A line that is perpendicular to a line segment and passes through its midpoint.

Circle

The locus of all points in a plane that are a fixed distance, known as the radius, from a given point known as the centre.

Equation of a circle (Completed Square Form)

The equation $(x - a)^2 + (y - b)^2 = r^2$, where $(a, b)$ is the centre of the circle and $r$ is the radius.

Equation of a circle (Expanded General Form)

The equation $x^2 + y^2 + 2gx + 2fy + c = 0$, where the centre is $(-g, -f)$ and the radius is $t{\sqrt{g^2 + f^2 - c}}$ .

Angle in a semicircle

A geometric fact stating that the angle subtended by a diameter at the circumference of a circle is always a right angle ($90^\circ$).

Chord property

The property stating that the perpendicular from the centre of a circle to a chord bisects the chord.

Tangent property

The property stating that the tangent to a circle at a point is perpendicular to the radius at that point.

Intersection Discriminant (Line and Curve)

For the quadratic resulting from simultaneous equations $ax^2 + bx + c = 0$, the value $b^2 - 4ac$ determines intersection: >0 (two points), $=0$ (tangent), or <0 (no intersection).