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Bihar Board Class 8 Sanskrit Book Solutions अमृता Amrita Bhag 3

April 13, 2025

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Bihar Board Class 8th Maths Solutions गणित

April 13, 2025

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Bihar Board Class 8th Science Solutions विज्ञान

April 13, 2025

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Bihar Board Class 7th Science Solutions विज्ञान

April 13, 2025

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Bihar Board Class 7 English Book Solutions Radiance Part 2

April 13, 2025

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Bihar Board Class 7 Sanskrit Book Solutions अमृता Amrita Bhag 2

April 13, 2025

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Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4

April 13, 2025

Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Text Book Questions and Answers.

BSEB Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4

Bihar Board Class 10 Maths निर्देशांक ज्यामिति Ex 7.4

प्रश्न 1.
बिन्दुओं A (2, -2) और B(3, 7) को जोड़ने वाले रेखाखण्ड को रेखा 2x + y – 4 = 0 जिस अनुपात में विभाजित करती है, उसे ज्ञात कीजिए।
हल
दिया है, बिन्दु A = (2, -2) तथा B = (3, 7)
यहाँ x₁ = 2, y₁ = -2, x₂ = 3, y₂ = 7
माना दिए हुए बिन्दुओं से बना रेखाखण्ड रेखा 2x + y – 4 = 0 को m₁ : m₂ के अनुपात में विभाजित करता है जबकि प्रतिच्छेद बिन्दु (x, y) है।
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q1
बिन्दु (x, 3) रेखा 2x + y – 4 = 0 पर स्थित होगा;
अतः इसके निर्देशांक रेखा 2x + y – 4 = 0 को सन्तुष्ट करेंगे।
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q1.1
अत: अभीष्ट अनुपात = 2 : 9

प्रश्न 2.
x और y में एक सम्बन्ध ज्ञात कीजिए यदि बिन्दु (x, y), (1, 2) और (7, 0) संरेखी है।
हल
माना बिन्दु A = (x, y), B = (1, 2) तथा C = (7, 0)
यहाँ, x₁ = x, y₁ = y, x₂ = 1, y₂ = 2, x₃ = 7, y₃ = 0
∆ का क्षेत्रफल = \(\frac{1}{2}\) [{x₁y₂ + x₂y₃ + x₃y₁} – {y₁x₂ + y₂x₃ + y₃x₁}]
= \(\frac{1}{2}\) [{x × 2 + 1 × 0 + 7 × y} – {y × 1 + 2 × 7 + 0 × x}]
= \(\frac{1}{2}\) [{2x + 0 + 7y} – {y + 14 + 0}]
= \(\frac{1}{2}\) [2x + 7y – y – 14]
= \(\frac{1}{2}\) [2x + 6y – 14]
= \(\frac{2}{2}\) (x + 3y – 7)
= x + 3y – 7
परन्तु यदि बिन्दु A, B, C संरेख हों तो ΔABC का क्षेत्रफल शून्य होना चाहिए।
x + 3y – 7 = 0
अतः x और में सम्बन्ध : x + 3y – 7 = 0

प्रश्न 3.
बिन्दुओं (6, -6), (3, -7) और (3, 3) से होकर जाने वाले वृत्त का केन्द्र ज्ञात कीजिए।
हल
माना A(6, -6), B(3, -7) तथा C(3, 3) बिन्दु एक वृत्त की परिधि पर हैं और वृत्त का केन्द्र O(h, k) है।
तब, OA, OB तथा OC वृत्त की त्रिज्याएँ होंगी।
अतः OA = OB = OC
⇒ OA² = OB² = OC²
OA² = [ केन्द्र O(h, k) और बिन्दु A (6, -6) के बीच की दूरी]²
⇒ OA² = (h – 6)² + (k + 6)²
⇒ OA² = h² – 12h + 36 + k² + 12k + 36
⇒ OA² = h² + k² – 12h + 12k + 72 ……..(1)
OB² = [केन्द्र O (h, k) और बिन्दु B (3, -7) के बीच की दूरी]²
⇒ OB² = (h – 3)² + (k + 7)²
⇒ OB² = h² – 6h + 9 + k² + 14k + 49
⇒ OB² = h² + k² – 6h + 14k + 58 ………(2)
OC² = [केन्द्र O(h, k) और बिन्दु C(3, 3) की दूरी]²
⇒ OC² = (h – 3)² + (k – 3)²
⇒ OC² = h² – 6h + 9 + k² – 6k + 9
⇒ OC² = h² + k² – 6h – 6k + 18 ………(3)
समीकरण (2) में से समीकरण (3) को घटाने पर,
20k + 40 = OB² – OC² = 0
⇒ k = -2
समीकरण (1) में से समीकरण (2) को घटाने पर,
-6h – 2k + 14 = OA² – OB² = 0
⇒ 6h + 2k = 14
⇒ 6h + (2 × -2) = 14 (∵ k = -2)
⇒ 6h – 4 = 14
⇒ 6h = 14 + 4 = 18
⇒ h = 3
अत: वृत्त का केन्द्र = (3, -2)

प्रश्न 4.
किसी वर्ग के दो सम्मुख शीर्ष (-1, 2) और (3, 2) हैं। वर्ग के अन्य दोनों शीर्ष ज्ञात कीजिए।
हल
दिया है, वर्ग के दो सम्मुख शीर्ष (-1, 2) व (3, 2) हैं।
वर्ग के एक विकर्ण का मध्य-बिन्दु = \(\left(\frac{-1+3}{2}, \frac{2+2}{2}\right)\) = (1, 2)
वर्ग के विकर्ण की लम्बाई = \(\sqrt{(-1-3)^{2}+(2-2)^{2}}\)
= \(\sqrt{(-4)^{2}+0}\)
= √16
= 4 मात्रक
तब, विकर्णों के प्रतिच्छेद बिन्दु E(मध्य बिन्दु) से प्रत्येक शीर्ष विकर्ण × \(\frac {1}{2}\) = 2 मात्रक दूरी पर होगा।
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q4
चित्र से स्पष्ट है कि शेष दोनों बिन्दु विकर्ण BD पर होंगे जो AC पर लम्ब होगा। तब प्रत्येक बिन्दु का भुज +1 होगा। माना कोटि y है।
तब, बिन्दु (1, 2) की बिन्दु (+1, y) से दूरी = 2 मात्रक
\(\sqrt{(1-1)^{2}+(y-2)^{2}}=2\)
⇒ \(\sqrt{0+(y-2)^{2}}=2\)
⇒ ±(y – 2) = 2
⇒ y – 2 = ±2
⇒ y = ±2 + 2
⇒ y = 0 या 4
अत: वर्ग के शेष दोनों शीर्ष (1, 0) व (1, 4) हैं।

प्रश्न 5.
कृष्णानगर के एक सेकेण्डरी स्कूल के कक्षा X के विद्यार्थियों को उनके बागवानी क्रियाकलाप के लिए एक आयताकार भूखण्ड दिया गया है। गुलमोहर की पौध (sapling) को परस्पर 1 मीटर की दूरी पर इस भूखण्ड की परिसीमा (boundary) पर लगाया जाता है। इस भूखण्ड के अन्दर एक त्रिभुजाकार घास लगा हुआ लॉन (lawn) है, जैसा कि आकृति में दर्शाया गया है। विद्यार्थियों को भूखण्ड के शेष भाग में फूलों के पौधे के बीज बोने हैं।
(i) A को मूलबिन्दु मानते हुए, त्रिभुज के शीर्षों के निर्देशांक ज्ञात कीजिए।
(ii) यदि मूलबिन्दु C हो तो ∆PQR के शीर्षों के निर्देशांक क्या होंगे?
साथ ही उपर्युक्त दोनों स्थितियों में, त्रिभुजों के क्षेत्रफल ज्ञात कीजिए। आप क्या देखते हैं?
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q5
हल
बिन्दुओं P, Q व R से सम्मुख अक्षों पर लम्ब खींचे गए हैं। (चित्र देखिए)

(i) यदि A मूलबिन्दु हो तो
बिन्दु P = (4, 6),Q = (3, 2) तथा R = (6, 5)
यहाँ x₁ = 4, y₁ = 6, x₂ = 3, y₂ = 2, x₃ = 6, y₃ = 5
∆PQR का क्षेत्रफल = \(\frac{1}{2}\) [{x₁y₂ + x₂y₃ + x₃y₁} – {y₁x₂ + y₂x₃ + y₃x₁}]
= \(\frac{1}{2}\) {{4 × 2 + 3 × 5 + 6 × 6} – {6 × 3 + 2 × 6 + 5 × 4}]
= \(\frac{1}{2}\) [(8 + 15 + 36) – (18 + 12 + 20)]
= \(\frac{1}{2}\) [59 – 50]
= \(\frac{1}{2}\) × 9
= \(\frac{9}{2}\) वर्ग मात्रक

(ii) जब C मूलबिन्दु हो तो
बिन्दु P = (-12, -2), Q = (-13, -6) तथा R = (-10, -3)
यहाँ x₁ = -12, y₁ = -2, x₂ = -13, y₂ = -6, x₃ = -10, y₃ = -3
∆PQR का क्षेत्रफल = \(\frac{1}{2}\) [{x₁y₂ + x₂y₃ + x₃y₁} – {y₁x₂ + y₂x₃ + y₃x₁}]
= \(\frac{1}{2}\) [{(-12 × -6) + (-13 × -3) + (-10 × -2)} – {(-2 × -13) + (-6 × -10) + (-3 × -12)}]
= \(\frac{1}{2}\) (72 + 39 + 20) – (26 + 60 + 36)]
= \(\frac{1}{2}\) [(131) – (122)]
= \(\frac{1}{2}\) × 9
= \(\frac{9}{2}\) वर्ग मात्रक
अत: दोनों ही स्थितियों में त्रिभुज का क्षेत्रफल समान है।

प्रश्न 6.
एक त्रिभुज ABC के शीर्ष A (4, 6), B(1, 5) और C (7, 2) हैं। भुजाओं AB और AC को क्रमशः D और E पर प्रतिच्छेद करते हुए एक रेखा इस प्रकार खींची गई है कि \(\frac{A D}{A B}=\frac{A E}{A C}=\frac{1}{4}\) है। ΔADE का क्षेत्रफल परिकलित कीजिए और इसकी तुलना ΔABC के क्षेत्रफल से कीजिए।
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q6
हल
दिया है, ΔABC के शीर्ष A (4, 6), B(1, 5) और C (7, 2) हैं।
\(\frac{A D}{A B}=\frac{1}{4}\)
⇒ AB = 4AD
⇒ AD + DB = 4AD
⇒ DB = 3AD
⇒ \(\frac{A D}{D B}=\frac{1}{3}\)
माना D के निर्देशांक यदि (x, y) हों तो
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q6.1

प्रश्न 7.
मान लीजिए A(4, 2), B(6, 5) और C (1, 4)एक त्रिभुज ABC के शीर्ष हैं।
(i) A से होकर जाने वाली माध्यिका BC से D पर मिलती है। बिन्दु D के निर्देशांक ज्ञात कीजिए।
(ii) AD पर स्थित ऐसे बिन्दु P के निर्देशांक ज्ञात कीजिए कि AP : PD = 2 : 1 हो।
(iii) माध्यिकाओं BE और CF पर ऐसे बिन्दुओं Q और R के निर्देशांक ज्ञात कीजिए कि BQ : QE = 2 : 1 हो और CR : RF = 2 : 1 हो।
(iv) आप क्या देखते हैं?
[नोट – वह बिन्दु जो तीनों माध्यिकाओं में सार्वनिष्ठ हो, उस त्रिभुज का केन्द्रक (centroid) कहलाता है और यह प्रत्येक माध्यिका को 2 : 1 के अनुपात में विभाजित करता है।]
(v) यदि A(x₁, y₁), B(x₂, y₂) और C(x₃, y₃) त्रिभुज ABC के शीर्ष हैं तो इस त्रिभुज के केन्द्रक के निर्देशांक ज्ञात कीजिए।
हल
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q7

प्रश्न 8.
बिन्दुओं A(-1, -1), B(-1, 4), C(5, 4) और D(5, -1) से एक आयत ABCD बनता है। P, Q, R और S क्रमश: भुजाओं AB, BC, CD और DA के मध्य-बिन्दु हैं। क्या चतुर्भुज PQRS एक वर्ग है? क्या यह एक आयत है? क्या यह एक समचतुर्भुज है? सकारण उत्तर दीजिए।
हल
दिए हुए बिन्दु A = (-1, -1), B = (-1, 4), C = (5, 4) और D = (5, -1)
Bihar Board Class 10 Maths Solutions Chapter 7 निर्देशांक ज्यामिति Ex 7.4 Q8

∵ चतुर्भुज PQRS में, PQ = QR = RS = SP और विकर्ण PR ≠ विकर्ण QS
अत: चतुर्भुज PQRS एक समचतुर्भुज है।

Bihar Board Class 6 Hindi Book Solutions किसलय Kislay Bhag 1

April 10, 2025

Top academic experts at BiharBoardSolutions.com have designed BSTBPC BSEB Bihar Board Class 6 Hindi Book Solutions किसलय भाग 1 हिंदी बुक बिहार क्लास 6, Kislay Bhag 1 Hindi Book Bihar Class 6 Solutions PDF Free Download are part of Bihar Board Class 6th Solutions based on the latest NCERT syllabus.

Here we have updated the detailed SCERT Bihar Board 6th Class Hindi Book Solutions of किसलय भाग 1 BTBC Book Class 6 Hindi Solutions Answers Guide, Bihar Text Book Class 6 Hindi Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material to help students in understanding the concepts behind each question in a simpler and detailed way.

BSEB Bihar Board Class 6th Hindi Book Solutions Kislay Bhag 1

Kislay Hindi Book Class 6 Solutions Bihar Board

Kislay Hindi Book Bihar Class 6 Solutions

Bihar Board Class 6th Hindi व्याकरण

व्याकरण

Bihar Board Class 6th Hindi रचना

We hope that BSTBPC BSEB Bihar Board Class 6th Hindi Book Solutions किसलय भाग 1 हिंदी बुक बिहार क्लास 6, Kislay Bhag 1 Hindi Book Bihar Class 6 Solutions PDF Free Download of BTBC Book Class 6 Hindi Solutions Answers Guide, Bihar Text Book Class 6 Hindi Kislay Bhag 1 Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material will help students can prepare all the concepts covered in the syllabus.

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Bihar Board Class 6th Science Solutions विज्ञान

April 10, 2025

Top academic experts at BiharBoardSolutions.com have designed BSTBPC BSEB Bihar Board Class 6 Science Book Solutions विज्ञान PDF Free Download in Hindi Medium and English Medium are part of Bihar Board Class 6th Solutions based on the latest NCERT syllabus.

Here we have updated the detailed SCERT Bihar Board 6th Class Science Book Solutions Vigyan of BTBC Book Class 6 Science Solutions Answers Guide, Bihar Text Book Class 6 Science Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material to help students in understanding the concepts behind each question in a simpler and detailed way.

BSEB Bihar Board Class 6th Science Book Solutions विज्ञान

We hope that BSTBPC BSEB Bihar Board Class 6th Science Book Solutions विज्ञान PDF Free Download in Hindi Medium and English Medium of BTBC Book Class 6 Science Solutions Answers Guide, Bihar Text Book Class 6 Science Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material will help students can prepare all the concepts covered in the syllabus.

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Bihar Board Class 6th Maths Solutions गणित

April 10, 2025

Top academic experts at BiharBoardSolutions.com have designed BSTBPC BSEB Bihar Board Class 6 Maths Book Solutions गणित PDF Free Download in Hindi Medium and English Medium are part of Bihar Board Class 6th Solutions based on the latest NCERT syllabus.

Here we have updated the detailed SCERT Bihar Board 6th Class Maths Book Solutions Ganit of BTBC Book Class 6 Maths Solutions Answers Guide, Bihar Text Book Class 6 Maths Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material to help students in understanding the concepts behind each question in a simpler and detailed way.

BSEB Bihar Board Class 6th Maths Book Solutions गणित

We hope that BSTBPC BSEB Bihar Board Class 6th Maths Book Solutions गणित PDF Free Download in Hindi Medium and English Medium of BTBC Book Class 6 Maths Solutions Answers Guide, Bihar Text Book Class 6 Maths Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material will help students can prepare all the concepts covered in the syllabus.

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Bihar Board Class 6 Sanskrit Book Solutions अमृता Amrita Bhag 1

April 10, 2025

Top academic experts at BiharBoardSolutions.com have designed BSTBPC BSEB Bihar Board Class 6 Sanskrit Book Solutions अमृता भाग 1 संस्कृत बुक बिहार क्लास 6, Amrita Bhag 1 Sanskrit Book Bihar Class 6 Solutions PDF Free Download are part of Bihar Board Class 6th Solutions based on the latest NCERT syllabus.

Here we have updated the detailed SCERT Bihar Board 6th Class Sanskrit Book Solutions अमृता भाग 1 of BTBC Book Class 6 Sanskrit Solutions Answers Guide, Bihar Text Book Class 6 Sanskrit Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material to help students in understanding the concepts behind each question in a simpler and detailed way.

BSEB Bihar Board Class 6th Sanskrit Book Solutions Amrita Bhag 1

Amrita Sanskrit Book Class 6 Solutions Bihar Board

Amrita Sanskrit Book Bihar Class 6 Solutions

Bihar Board Class 6th Sanskrit व्याकरण

We hope that BSTBPC BSEB Bihar Board Class 6th Sanskrit Book Solutions अमृता भाग 1 संस्कृत बुक बिहार क्लास 6, Amrita Bhag 1 Sanskrit Book Bihar Class 6 Solutions PDF Free Download of BTBC Book Class 6 Sanskrit Solutions Answers Guide, Bihar Text Book Class 6 Sanskrit Questions and Answers, Chapter Wise Notes Pdf, Model Question Papers, Study Material will help students can prepare all the concepts covered in the syllabus.

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