Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach combines algebra and geometry to describe the position of points, lines, and curves.
Coordinate System: The coordinate system is a method to identify the position of a point in a plane by using two numbers, called coordinates. The most common system is the Cartesian coordinate system, where the position of a point is determined by its distance from two perpendicular axes intersecting at a point called the origin.
Points: A point in coordinate geometry is represented by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the y-axis, and \(y\) is the vertical distance from the x-axis.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a plane is given by the formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) This formula is derived from the Pythagorean theorem applied to the right triangle formed by the line connecting the two points and the projections of this line onto the x-axis and y-axis.
The midpoint of the line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the following formula: \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \) The midpoint is the point that divides the line segment into two equal parts.
Slope-Intercept Form: The equation of a straight line in the slope-intercept form is \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. The slope \(m\) represents the steepness of the line and is calculated as the change in y over the change in x between two points on the line.
Point-Slope Form: Another form of the equation of a line is the point-slope form, which is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope of the line.
The equation of a circle with center \((h, k)\) and radius \(r\) is given by: \( (x - h)^2 + (y - k)^2 = r^2 \) This equation represents all points \((x, y)\) that are a distance \(r\) from the center \((h, k)\).
A parabola is a curve where any point is at an equal distance from a fixed point called the focus and a fixed line called the directrix. The standard form of the equation of a parabola opening upwards or downwards is: \( y - k = a(x - h)^2 \) Where \((h, k)\) is the vertex of the parabola, and \(a\) is a coefficient that determines the width and direction of the parabola.
Example 1: Calculate the distance between points (2, 3) and (-1, -1). By applying the distance formula, we have: \( d = \sqrt{(-1 - 2)^2 + (-1 - 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Example 2: Find the midpoint of the line segment connecting points (6, 4) and (2, -2). Using the midpoint formula, we get: \( M = \left(\frac{6 + 2}{2}, \frac{4 - 2}{2}\right) = (4, 1) \)
Example 3: Write the equation of the line with slope 2 passing through the point (3, -1). Using the point-slope form, we have: \( y - (-1) = 2(x - 3) \) Simplifying, we obtain: \( y = 2x - 7 \)
To further understand coordinate geometry, it is helpful to use graphing software to visualize the equations and concepts discussed. By inputting different equations, one can see how changing values affect the shape and position of the geometric figures.
Coordinate geometry is a powerful tool that allows us to precisely describe the position and characteristics of geometric figures in a plane. It bridges algebra and geometry, providing a way to analyze and solve geometric problems through algebraic equations.
