COORDINATE GEOMETRY & STATISTICS

Bihar STET Paper 1 - Coordinate Geometry & Statistics

निर्देशांक ज्यामिति एवं सांख्यिकी • All Repeated PYQ Types + Visuals (Pg 127–146 & 250–282)

Weightage: 12–15 Marks Direct Exam Hacks & Visuals 100% Blogger Ready

Type 1: Distance Formula & Origin Distance (दूरी सूत्र)

Formula #1

d = √[(x2 - x1)² + (y2 - y1)²]  |  Origin se doori = √(x² + y²)

⚡ Pythagorean Triplet Trick: Δx aur Δy ka difference nikalo. Agar difference (3, 4) dikhe toh distance seedhe 5, agar (5, 12) ho toh seedhe 13, agar (7, 24) ho toh seedhe 25!

[PYQ Example] Point P(-6, 8) ki Origin (मूल बिंदु) se distance kya hogi?

d = √[(-6)² + 8²] = √[36 + 64] = √100 = 10 units
(Triplet: 6, 8 ➔ 10)
✔ Answer: 10

Type 2: Section Formula, Midpoint & Axis Ratio Division (विभाजन सूत्र)

Most Repeated

x = (m1·x2 + m2·x1) / (m1 + m2)  |  y = (m1·y2 + m2·y1) / (m1 + m2)

⚡ Super Direct Exam Hacks (Bina Formula Lagaye):
• Midpoint (मध्य बिंदु): [(x1 + x2) / 2 , (y1 + y2) / 2]
• X-axis kis ratio me baant-ti hai? Direct Ratio = -y1 : y2
• Y-axis kis ratio me baant-ti hai? Direct Ratio = -x1 : x2

[PYQ Example] Points (2, -3) aur (5, 6) ko milane wali line ko X-axis kis ratio me divide karti hai?

Direct Hack for X-axis: Ratio = -y1 / y2
Yahan y1 = -3, y2 = 6
Ratio = -(-3) / 6 = 3 / 6 = 1 : 2
✔ Answer: 1 : 2

Type 3: Collinearity (संरेखीय बिंदु) & Centroid of Triangle

Zero Area Trick

Core Formulas:
• Collinear Points (संरेख बिंदु): Teen points ek hi line par honge toh unse bane Triangle ka Area = 0 hoga!
• Centroid of Δ (केंद्रक G): G = [(x1 + x2 + x3)/3 , (y1 + y2 + y3)/3]

[PYQ Example] If points (2, 3), (4, k) and (6, -3) are collinear, find the value of k:

Collinear ➔ Slope 1 = Slope 2 (Sabse fast shortcut!)
(k - 3) / (4 - 2) = (-3 - k) / (6 - 4)
(k - 3) / 2 = (-3 - k) / 2 ⇒ k - 3 = -3 - k
2k = 0 ⇒ k = 0
✔ Answer: k = 0

Type 4: Empirical Relation (माध्य, माध्यिका एवं बहुलक का संबंध)

100% Fixed Question

Mode = 3 Median - 2 Mean

बहुलक = 3 × माध्यिका - 2 × माध्य

[PYQ Example - Bihar STET 2024 Shift-I] If Mean = 24 and Mode = 12, find Median:

Formula: Mode = 3 × Median - 2 × Mean
12 = 3 × Median - 2 × 24
12 = 3 × Median - 48 ⇒ 3 × Median = 12 + 48 = 60
Median = 60 / 3 = 20
✔ Answer: 20

Type 5: Ogive Curve (तोरण) & Properties of Mean/Variance

Theory & Concept

'Less than' aur 'More than' Ogive ka intersection point hamesha Median (माध्यिका) deta hai!

Theory PYQ Points:
• Histogram (आयतचित्र): Isse Mode (बहुलक) graphically find kiya jata hai.
• Cumulative Frequency Curve (Ogive / तोरण): Isse Median (माध्यिका) milti hai.
• Mean Property: Agar har observation me 'k' add/multiply kar diya jaye, toh naya Mean bhi puraane mean me 'k' add/multiply ho jayega!

[PYQ Theory] The point of intersection of 'less than' and 'more than' ogives corresponds to:

(a) Mean    (b) Mode    (c) Median    (d) None
✔ Correct Answer: (c) Median (माध्यिका)