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Revision Guide
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Coordinate Geometry - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Coordinate Geometry aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Key Points
Define coordinate axes.
Coordinate axes are two perpendicular lines, horizontal (x-axis) and vertical (y-axis), used to locate points on a plane. The point of intersection is called the origin.
What are abscissa and ordinate?
Abscissa is the x-coordinate (distance from y-axis), and ordinate is the y-coordinate (distance from x-axis) of a point.
Distance formula between two points.
The distance between points (x1, y1) and (x2, y2) is √[(x2-x1)² + (y2-y1)²]. It's derived from Pythagoras theorem.
Mid-point formula.
The mid-point of a segment joining (x1, y1) and (x2, y2) is [(x1+x2)/2, (y1+y2)/2]. It's the average of x and y coordinates.
Section formula (internal division).
A point dividing the line joining (x1, y1) and (x2, y2) in ratio m:n is [(mx2+nx1)/(m+n), (my2+ny1)/(m+n)].
How to check collinearity of three points?
Three points are collinear if the area formed by them is zero or if the slope between any two pairs is the same.
Area of a triangle using coordinates.
Area = 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|. Absolute value ensures the area is non-negative.
Slope of a line.
Slope (m) = (y2-y1)/(x2-x1). It measures the steepness and direction of a line. Undefined for vertical lines.
Condition for parallel lines.
Two lines are parallel if their slopes are equal. Parallel lines never meet.
Condition for perpendicular lines.
Two lines are perpendicular if the product of their slopes is -1. They intersect at right angles.
Equation of x-axis and y-axis.
x-axis: y=0; y-axis: x=0. These are the simplest forms of linear equations.
Equation of a line parallel to x-axis.
y = k, where k is a constant. All points on this line have the same y-coordinate.
Equation of a line parallel to y-axis.
x = k, where k is a constant. All points on this line have the same x-coordinate.
Centroid of a triangle.
The intersection point of medians. Coordinates are the average of the vertices' coordinates: [(x1+x2+x3)/3, (y1+y2+y3)/3].
How to find the ratio in which a point divides a line?
Use section formula inversely. Compare given point's coordinates with section formula to find m:n.
Real-world application of coordinate geometry.
Used in GPS navigation, computer graphics, and engineering designs to map locations and shapes accurately.
Misconception: Distance formula is only for positive coordinates.
Distance formula works for all real numbers. Negative coordinates represent positions in different quadrants.
Memory hack for area of triangle formula.
Remember as half of |x1y2 + x2y3 + x3y1 - x1y3 - x2y1 - x3y2|. It's cyclic and easy to recall.
Important note on slope.
Slope is a measure of rate of change. A positive slope means the line rises, negative means it falls.
Quick check for collinearity.
If the sum of any two distances equals the third, points are collinear. Faster than area method.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
