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MTE 5: Analytical Geometry

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Title Name	IGNOU MTE 5 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 5
Subject Name	Analytical Geometry
Year	2025
Session	-
Language	English Medium
Assignment Code	MTE 5/Assignment-1/2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE 05 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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MTE 5 2025 - English

Assignment

Course Code: MTE-05

Assignment Code: MTE-05/TMA/2025

Maximum Marks: 100

1. Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.

(i) The numbers $frac{1}{sqrt{2}},frac{1}{sqrt{3}},frac{3}{4}$ are the direction cosines of a line.

(ii) The points (1, 2), (7, 6) and (4, 4) are collinear.

(iii) The conic 12x² + 12xy + 3y² + 2x + y = 0 is degenerate.

(iv) Intersection of the ellipsoid x²/4+ y² /25+ z² /4 = 1 and the plane y = 5 is a circle.

(v) The conicoid 3x² + y² + 2xy+x-y-z+1=0 is non-central.

(vi) The line y = x is a tangent to the parabola y² = cx, c > 0.

(vii) The equation 2x² + y + z + 1 = 0 represents a paraboloid.

(viii) Projection of a line segment on a line perpendicular to it is the length of the line segment.

(ix) The lines x =- y, z = 2 and x = y, z = 0 intersect each other.

(x) Every planar section of a cylinder is a circle.

2. (a) Trace the conic x² 2xy + y² 3x + 2y + 3 = 0.

(b) Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

(a) Let P be the midpoint of the line segment joining the points A(a + b, b) and B(a - b, a + b). Find the slope of the line passing through P and Q (b,- a/2). Under what conditions on a and b, this line is parallel to the y-axis?

(b) i) Show that $egin{vmatrix}x&y&1\2&3&1\-4&7&1end{vmatrix}=0$ represents the equation of a line passing 1-4 7 1 through (2, 3) and (-4, 7).

(ii) Prove that the equation of a line through (x1, y1) and (x2, y2) can be expressed in the form

$egin{vmatrix}x&y&1\x_1&y_1&1\x_2&y_2&1end{vmatrix}=0$

(c) Find the eccentricity, foci, centre and directrices of the ellipse $frac{x^2}{16}+frac{y^2}{4}=1$ . Also 4 give a rough sketch of it.

(d) Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.

(a) Find the equations of the line through (1,3,4) and parallel to the line joining the points (-4, 5, 3) and (8, 9, 7).

(b) Find the equation of the plane which passes through the line of intersection of the planes 3x + 4y - 5z = 9 and 2x+6y+6z = 7 and which is perpendicular to the plane 3x + 2y5z + 6 = 0.

(a) Show that the plane 2x + y + 2z = 0 is a tangent plane to the sphere x² + y² + z2-2x+2y-2z + 2 = 0.

(b) Find the equation of the sphere touching the plane 8x + 5y + 3z + 1 = 0 at (3,-1,-1) and cutting the sphere x² + y2 + z²-2x+y-z-6=0 orthogonally.

(c) Find the angle between the lines of intersection of the cone 4x2 + y² + 4z² + 4yz + 2zx = 0 and the plane x + 2y + 3z = 0.

(d) Find the equation of the cylinder with base x² + y² + z²-3x6z + 9 = 0, x - 2y+2z-6 = 0.

(a) Show that the perpendiculars drawn from the origin to tangent planes to the cone x2 y2 + 5z² + 4xy = 0 lie on the cone x2 y2 + z² + 4xy = 0.

(b) Transform the equation x² + 2y² 6z²- 2x - 8y+3 = 0 by shifting the origin to (1, 2, 0) without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.

(a) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

(i) x² + y² + z² + 4x + 3y – z = 0

(ii) 2x2-y2z2 + xy + yz - zx = 1

(iii) x2 + y2 - z2 -2xy -3yz - 6zx + x - 2y + 5z + 4 = 0

(b) Find the transformation of the equation 12x² - 2y2 + z² = 2xy if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1, -3, 0; 3, 1, 0; 0, 0, 1.

8. (a) Identify and trace the conicoid y² + 3z2 = x. Describe its sections by the planes y = 0 and z = 0.

(b) Find the equation of tangent plane to the conicoid x² + 3y² = 4z at (2, -4, 13). Represent the tangent plane geometrically.

MTE 05 (January 2025 - July 2025) - HINDI

सत्रीय कार्य

पाठ्यक्रम कोड: MTE-05

सत्रीय कार्य कोड: MTE-05/TMA/2025

अधिकतम अंकः 100

1. जांच कीजिए कि निम्नलिखित कथन सत्य हैं अथवा असत्य। अपने उत्तर की पुष्टि लघु व्याख्या या प्रति-उदाहरण द्वारा कीजिए।

(i) संख्याएँ $frac{1}{sqrt{2}},frac{1}{sqrt{3}},frac{3}{4},dots$ एक रेखा की दिक्कोज्याएं हैं।

(ii) बिंदु (1, 2), (7,6) और (4,4) संरेखीय हैं।

(iii) शंकव $12x^2+12xy+3y^2+2x+y=0$ अपभ्रष्ट है।

(iv) दीर्घवृत्त $frac{x^2}{4}+frac{y^2}{25}+frac{z^2}{4}=1$ का समतल y = 5 से प्रतिच्छेद एक वृत्त है।

(v) शांकवज $3x^2+y^2+2xy+x-y-z+1=0$ अकेंद्रीय है।

(vi) रेखा y = x परवलय $y^2=cx,c>0$ की स्पर्श रेखा है।

(vii) समीकरण $2x^2+y+z+1=0$ एक परवलज को निरूपित करता है।

(viii) किसी रेखा-खण्ड का उसकी लंब रेखा पर प्रक्षेप उस रेखा-खण्ड की लंबाई के बराबर होता है।

(ix) रेखाएं $x=-y,z=2$ और $x=y,z=0$ परस्पर प्रतिच्छेद करती हैं।

(x) बेलन का समतल परिच्छेद एक वृत्त होता है।

2.(क) शांकव $x^2-2xy+y^2-3x+2y+3=0$ को अनुरेखित कीजिए।

(ख) सिद्ध कि दो समकोणीय अतिपरवलयों के प्रतिच्छेद बिन्दुओं से अंतिम वाला शंकव भी समकोणीय अतिपरवल होता है।

(ग) दिखाइए कि रेखा x = y शांकव $ax^2+2hxy+by^2+2gx+2fy+c=0$ को स्पर्श करेगी, यदि $f+g=0$ हो।

3.(क) मान लीजिए P बिंदुओं A(a + b, b) और B(ab, a + b) को मिलाने वाले रेखा-खण्ड का मध्य-बिंदु है। P और Q (b) से गुजरने वाली रेखा की प्रवणता निकालिए। a और b पर किन प्रतिबंधों के अधीन यह रेखा -अक्ष के समांतर होगी?

(ख) (i) दिखाइए कि $egin{vmatrix}x&y&1\2&3&1\-4&7&1end{vmatrix}=0$ बिंदुओं (2,3) और (-4,7) से गुजरने वाली रेखा को निरूपित करता है।

(ii) सिद्ध कीजिए कि (x₁,y₁) और (x_2, y₂) से गुजरने वाली रेखा के समीकरण y को $egin{vmatrix}x&y&1\x_1&y_1&1\x_2&y_2&1end{vmatrix}=0$ के रूप में लिखा जा सकता है।

(ग) दीर्घवृत्त $frac{x^2}{16}+frac{y^2}{4}=1$ की उत्केंद्रता, नाभियां, केंद्र और नियताएं ज्ञात कीजिए। इसका स्थूल चित्र भी बनाइए।

(घ) सिद्ध कीजिए कि किसी परवलय की नाभि से गुजरने वाली तथा उस परवलय की अक्ष से 30° पर झुकी हुई जीवा की लंबाई, उस परवलय की नामिलंव की लंबाई की चार गुना होती है।

4.(क) बिंदु (1,3,4) से गुजरने वाली तथा बिंदुओं (-4,5,3) और (8,9,7) को मिलाने वाली रेखा के समांतर रेखा के समीकरण ज्ञात कीजिए।

(ख) समतलों $3x+4y-5z=9 ext{}2x+6y+6z=7$ की प्रतिच्छेद रेखा से गुजरने वाले तथा समतल $3x+2y-5z+6=0$ पर लंब समतल का समीकरण ज्ञात कीजिए।

(ग) मूलबिंदु की उस समतल से दूरी ज्ञात कीजिए जो बिंदुओं (2, 1, 8 , 1, 0, 2) और (-3,4,6) से गुजरता है।

5.(क) दिखाइए कि समतल $2x+y+2z=0$ गोले $x^2+y^2+z^2-2x+2y-2z+2=0$ का स्पर्श तल है।

(ख) समतल $8x+5y+3z+1=0$ को (3,-1,-1) पर स्पर्श करने वाले तथा गोले $x^2+y^2+z^2-2x+y-z-6=0$ को लांबिकतः प्रतिच्छेद करने वाले गोले का समीकरण ज्ञात कीजिए।

(ग) शंकु $4x^2+y^2+4z^2+4yz+2zx=0$ तथा समतल $x+2y+3z=0$ की प्रतिच्छेदी रेखाओं के बीच का कोण ज्ञात कीजिए।

(घ) उस बेलन का समीकरण ज्ञात कीजिए जिसका आधार $x^2+y^2+z^2-3x-6z+9=0$ $x-2y+2z-6=0$ है।

6.(क) दिखाइए कि शंकु $x^2-y^2+5z^2+4xy=0$ के स्पर्श तों पर मूलबिंदु से डाले गए लंब शंकु $x^2-y^2+z^2+4xy=0$

(ख) निर्देशांक अक्षों की दिशाओं को परिवर्तित किए बिना मूलबिंदु को (1,2,0) पर स्थानांतरित करके समीकरण $x^2+2y^2-6z^2-2x-8y+3=0$ को रूपांतरित कीजिए। यह नया समीकरण क्या निरुपित करता है? इसका स्थूल आरेख बनाइए।

(ग) दिखाइए कि शांकवज $2x^2+2y^2+xy-yz+2x+2x-y+5z+1$ केंद्रीय है। अतः इसका केंद्र निकालिए।

7.(क) जांच कीजिए कि निम्नलिखित शांकवों में से कौनसे शांकवज केंद्रीय हैं और कौनसे अकेंद्रीय हैं। यह भी पता कीजिए कि जो केंद्रीय शांकवज हैं उनमें से किनके केंद्र मूलबिंदु पर है।

(i) $x^2+y^2+z^2+4x+3y-z=0$

(ii) $2x^2-y^2-z^2+xy+yz-zx=1$

(iii) $x^2+y^2-z^2-2xy-3yz-6zx+x-2y+5z+4=0$

(ख) समीकरण $12x^2-2y^2+z^2=2xy$ को रूपांतरित कीजिए, यदि मूलबिंदु को स्थिर रखा 2 जाए और अक्षों को इस प्रकार घुमाया जाए कि नए अक्षों के दिक-अनुपात 1,-3,0; 3, 1, 0, 0, 0,1 हो।

(ग) बिंदुओं (1,-1,6) और (4,3,2) को मिलाने वाले रेखा-खण्ड का रेखा $frac{x-4}{3}=-y=frac{z}{5}$ प्रक्षेप ज्ञात कीजिए।

8.(क) शांकवज $y^2+3z^2=x$ को पहचानिए तथा आरेखित कीजिए। समतलों y = 0 और z = 0 द्वारा इसके परिच्छेदों का वर्णन कीजिए।

(ख) बिंदु (2,-4,13) पर शांकवज $x^2+3y^2=4z$ के स्पर्श तल का समीकरण ज्ञात कीजिए। स्पर्श तल को ज्यामितीय रूप से दर्शाइए।

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