Orienting Yourself: The Use of Coordinates (Class 9): Notes और MCQ Practice

आसान भाषा में

Coordinate Geometry एक 2-D grid system है, जिससे हम plane में किसी भी point की exact location दो numbers (x, y) के ज़रिए दर्शाते हैं। इसमें horizontal line को x-axis और vertical line को y-axis कहते हैं, जो एक-दूसरे को Origin (0, 0) पर काटती हैं।

Important Points & Formulas

• Origin (O): Axes का intersection point, इसके coordinates (0, 0) होते हैं।
• Points on Axes:
• x-axis पर स्थित point: (x, 0)
• y-axis पर स्थित point: (0, y)
• Four Quadrants:
• Quadrant I: (+, +)
• Quadrant II: (−, +)
• Quadrant III: (−, −)
• Quadrant IV: (+, −)
• Ordered Pair: यदि x ≠ y, तो (x, y) ≠ (y, x)।
• Coordinates का अर्थ: x = y-axis से perpendicular दूरी; y = x-axis से perpendicular दूरी।
• Distance Formula (Baudhāyana–Pythagoras Theorem):

Points (x₁, y₁) और (x₂, y₂) के बीच की दूरी = √((x₂ − x₁)² + (y₂ − y₁)²)

2 Memory Tricks

1. "Walk, then Climb" (x पहले, y बाद में): पहले ज़मीन पर सीधे चलो (x-axis: left/right), फिर सीढ़ी चढ़ो या उतरो (y-axis: up/down)। हमेशा order (x, y) रहेगा।
2. "Anti-Clockwise Quadrants": Top-right से शुरू करके घड़ी की उल्टी दिशा (Anti-clockwise) में घूमें: Q1 (+, +) → Q2 (−, +) → Q3 (−, −) → Q4 (+, −)।

2 Common Exam Mistakes

1. Axis पर Coordinates उलटना: बच्चे अक्सर x-axis के point को (0, x) लिख देते हैं। याद रखें, x-axis पर हमेशा y = 0 होता है, यानी (x, 0)।
2. (x, y) और (y, x) को समान मानना: जैसे (3, −5) Quadrant IV में है, जबकि (−5, 3) Quadrant II में है। Order बदलते ही point बदल जाता है!

Orienting Yourself: The Use of Coordinates

1. Chapter in Short

This chapter introduces Coordinate Geometry, a mathematical system that uses pairs of numbers to describe the exact physical locations of points or objects on a flat surface (two-dimensional plane).

We learn how two perpendicular number lines—a horizontal x-axis and a vertical y-axis—divide a plane into four quadrants intersecting at the origin (0, 0). The chapter explores the rich history of grid-based thinking (from the ancient Sindhu-Sarasvatī Civilisation and Indian astronomers to René Descartes), teaches how to plot and identify points using coordinates (x, y), and demonstrates how to calculate the straight-line distance between any two points using the Baudhāyana–Pythagoras Theorem.

2. Key Concepts Explained Simply

The Grid and Coordinate System

Imagine a city where all streets run North–South and East–West. If you start from the city centre, you can tell someone how to reach any building by saying: "Go 4 blocks East, then 3 blocks North." A coordinate system works the same way using numbers on graph paper.

The Axes and Origin

• x-axis: The horizontal line running left to right.
• y-axis: The vertical line running up and down.
• Origin (O): The point where the two axes cross each other. Its coordinates are (0, 0).
• Directions:
• Moving Right from O along the x-axis = Positive
• Moving Left from O along the x-axis = Negative
• Moving Up from O along the y-axis = Positive
• Moving Down from O along the y-axis = Negative

Reading and Writing Coordinates (x, y)

Every point is represented as an ordered pair (x, y):

• The first number, x, is the perpendicular distance of the point from the y-axis (measured along the x-axis).
• The second number, y, is the perpendicular distance of the point from the x-axis (measured along the y-axis).
• Order matters: Point (2, 5) is not the same as (5, 2). In fact, (x, y) = (y, x) only when x = y.

Points on the Axes

• Any point lying on the x-axis has a y-coordinate of 0. Its coordinates are of the form (x, 0).
• Any point lying on the y-axis has an x-coordinate of 0. Its coordinates are of the form (0, y).

The Four Quadrants

The two axes divide the plane into four regions called quadrants (numbered anticlockwise):

• Quadrant I: Top-right region, where both coordinates are positive (+, +).
• Quadrant II: Top-left region, where x is negative and y is positive (−, +).
• Quadrant III: Bottom-left region, where both coordinates are negative (−, −).
• Quadrant IV: Bottom-right region, where x is positive and y is negative (+, −).

Distance Between Two Points

• If two points lie on a horizontal line (same y-coordinate): Distance = |x2 − x1|
• If two points lie on a vertical line (same x-coordinate): Distance = |y2 − y1|
• For any two points (x1, y1) and (x2, y2): We form a right-angled triangle where the horizontal change is (x2 − x1) and the vertical change is (y2 − y1). Using the Baudhāyana–Pythagoras Theorem:

$\text{Distance} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$

Written simply: Distance = √((x2 − x1)² + (y2 − y1)²)

3. Important Definitions, Formulas, Facts, and Dates

Historical Facts and Milestones

• Sindhu-Sarasvatī Civilisation: First systematic use of urban grids thousands of years ago, with streets built North–South and East–West at uniform intervals of about 10 metres.
• Baudhāyana (c. 800 CE): Used East–West and North–South lines in geometric constructions, developing the Baudhāyana–Pythagoras Theorem.
• Ujjayinī: Recognized as early as the 4th century BCE in early Siddhāntas as the central longitude meridian for world geographical measurements.
• Ptolemy (c. 150 BCE): Recorded latitudes and longitudes of thousands of locations, referring to Ujjayinī as 'Ozine'.
• Āryabhaṭa (c. 499 CE): Introduced sines instead of Greek chords and used Celestial Coordinates measured from the ecliptic (path of the sun).
• Brahmagupta (c. 628 CE): Formalised zero and negative numbers. Without these, the four-quadrant plane could not exist. His work was translated into Arabic as the Sindhind (where Ujjayinī was called 'Arin').
• Al-Bīrūnī (c. 1000 CE): Used Indian trigonometry to calculate city coordinates across Asia and perfected the astrolabe.
• Ömar Khayyām (c. 1100 CE): First to solve algebraic equations using coordinates in the plane.
• Fermat (1636 CE) and René Descartes (1637 CE): Formalised that any point in a 2D plane can be specified by two numbers measuring distances from two perpendicular axes.

Essential Definitions

• Cartesian Plane / Coordinate Plane / xy-plane: The 2D surface defined by the two perpendicular coordinate axes.
• Coordinate Axes: The horizontal line (x-axis) and vertical line (y-axis).
• Origin: The intersection point of the coordinate axes, denoted by O(0, 0).
• Coordinates: An ordered pair (x, y) specifying a point's location.

Key Formulas

• Coordinates of origin: (0, 0)
• Point on x-axis: (x, 0)
• Point on y-axis: (0, y)
• Distance formula: Distance = √((x2 − x1)² + (y2 − y1)²)
• Midpoint connection: If M(x_m, y_m) is the midpoint of A(x1, y1) and B(x2, y2), then x_m = (x1 + x2) ÷ 2 and y_m = (y1 + y2) ÷ 2.

4. Solved Examples

Example 1: Identifying Quadrants and Locations

Question: State which quadrant or axis each of the following points lies on:

(a) P(4, 5)

(b) Q(−3, 6)

(c) R(−2, −7)

(d) S(5, −4)

(e) T(0, −8)

(f) U(9, 0)

Answer:

• P(4, 5): Both coordinates are positive (+, +) → Quadrant I
• Q(−3, 6): x is negative, y is positive (−, +) → Quadrant II
• R(−2, −7): Both coordinates are negative (−, −) → Quadrant III
• S(5, −4): x is positive, y is negative (+, −) → Quadrant IV
• T(0, −8): x-coordinate is 0, y is negative → Negative y-axis
• U(9, 0): y-coordinate is 0, x is positive → Positive x-axis

Example 2: Finding the Distance Between Two Points

Question: Find the distance between the points A(3, 4) and D(7, 1).

Answer:

1. Let (x1, y1) = (3, 4) and (x2, y2) = (7, 1).
2. Change along the x-axis:

x2 − x1 = 7 − 3 = 4

1. Change along the y-axis:

y2 − y1 = 1 − 4 = −3

1. Apply the distance formula:

Distance = √((x2 − x1)² + (y2 − y1)²)

Distance = √(4² + (−3)²)

Distance = √(16 + 9) = √25 = 5 units

Example 3: Verifying Points on a Circle

Question: Show that the points A(1, −8), B(−4, 7), and C(−7, −4) lie on a circle whose centre is the origin O(0, 0). What is the radius of the circle?

Answer:

A point lies on a circle if its distance from the centre (the origin) is equal to the radius.

1. Distance OA from O(0, 0) to A(1, −8):

OA = √((1 − 0)² + (−8 − 0)²)

OA = √(1² + (−8)²) = √(1 + 64) = √65 units

1. Distance OB from O(0, 0) to B(−4, 7):

OB = √((−4 − 0)² + (7 − 0)²)

OB = √((−4)² + 7²) = √(16 + 49) = √65 units

1. Distance OC from O(0, 0) to C(−7, −4):

OC = √((−7 − 0)² + (−4 − 0)²)

OC = √((−7)² + (−4)²) = √(49 + 16) = √65 units

Since OA = OB = OC = √65, all three points lie on the same circle centred at O(0, 0).

Radius of the circle = √65 units.

Example 4: Finding an Endpoint from a Midpoint

Question: Point M(−7, 1) is the midpoint of the line segment joining point A(3, −4) and point B(x, y). Find the coordinates of B.

Answer:

The midpoint coordinates are the averages of the coordinates of the endpoints:

1. For the x-coordinate:

(3 + x) ÷ 2 = −7

3 + x = −7 × 2

3 + x = −14

x = −14 − 3 = −17

1. For the y-coordinate:

(−4 + y) ÷ 2 = 1

−4 + y = 1 × 2

−4 + y = 2

y = 2 + 4 = 6

Therefore, the coordinates of B are (−17, 6).

5. Common Mistakes and Quick Revision Points

• Writing y before x: Coordinates are always in alphabetical order: (x, y). Always read or count the horizontal shift first, then the vertical shift.
• Confusing Axis Coordinates:
• A point on the x-axis has y = 0, e.g., (5, 0).
• A point on the y-axis has x = 0, e.g., (0, 5).
• Don't mix them up! A point like (0, 3) lies on the vertical y-axis, not the x-axis.
• Mistakes with Negative Numbers in Subtraction: When subtracting a negative number in the distance formula, remember that subtracting a negative becomes addition:

x2 − x1 = −3 − (−7) = −3 + 7 = 4.

• Squaring Negative Numbers: The square of any real number is always positive or zero. For instance, (−3)² = 9, never −9.
• Signs in Quadrants:
• Quadrant I: (+, +)
• Quadrant II: (−, +)
• Quadrant III: (−, −)
• Quadrant IV: (+, −)
• Order Matters: Unless x = y, the points (x, y) and (y, x) are completely different locations on the coordinate plane.