Unit 1: Analytical Geometry

The slope of perpedicular lines are negative reciprocal.
The slopes of parallel lines are the same
The slope of a vertical line is undefined
The slope of a horizontal line is 0.
The general equation of a line in standard form is \(`ax+by+c=0`\), where \(`a,b,c \in \mathbb{Z}, a>0`\)
Radius: The distance from the centre of a circle to a point on the circumference of the cricle.
Diameter: the distance across a circle measured through the centre
Chord: a line segment joining two points on a curve
Circle: a set of points in the plane which are equidistant (same distance) from the centre

Distance Formula

The distance between points \(`A(x_1, y_1)`\) and \(`B(x_2, y_2)`\) in the cartesian plane is:

\(`d = \sqrt{x^2 + y^2}`\)

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

Identifying Types of Traingles

Triangle	Property
Equilateral	3 equal sides. Each angle is 60 degrees. Can’t be right angled
Isoceles	2 equal sides, 2 equal angles. May be right angled
Scalene	No equal sides. No equal angles. May be right angled

Pythagorean Theorem Relationships

Formula	Statement
\(`c^2 = a^2+b^2`\)	The triangle must be right angled
\(`c^2 < a^2 + b^2`\)	The triangle is acute
\(`c^2 > a^2 + b^2`\)	The triangle is obtuse

Equation Of A Circle With Centre \(`(0, 0)`\)

Let \(`P(x, y)`\) be any point on the circle, and \(`O`\) be the origin \(`(0, 0)`\).

Using Pythagorean Theorem,

\(`x^2+ y^2 = OP^2`\)

But, \(`OP = r`\)

\(`\therefore x^2 + y^2 = r^2`\) is the equation of a circle with centre \(`(0, 0)`\) and radius, \(`r`\).

Note: the coordinates of any point not on the cricle do not satisfy this equation

Semi-Cricle With Radius \(`r`\), And Centre \(`(0, 0)`\)

If we solve for \(`y`\) in the above equation \(`y = \pm \sqrt{r^2-x^2}`\) - \(`y = +\sqrt{r^2-x^2}`\) is the top half of the circle. - \(`y = -\sqrt{r^2-x^2}`\) is the bottom half of the circle

Equation Of A Circle With Centre \(`(x, y)`\)

Let \(`x_c, y_c`\) be the center

\(`(x - x_c)^2 + (y - y_c)^2 = r^2`\)

To get the center, just find a \(`x, y`\) such that \(`x - x_c = 0`\) and \(`y - y_c = 0`\)

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