Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.
(i) The numbers 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.
Let's break down each statement and solve it one by one. ### (i) The numbers \( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{3}{4} \) are the direction cosines of a line. False. The direction cosines of a line must satisfy the condition: \[ l^2 + m^2 + n^2 = 1 \] where \(l\), \(m\), and \(n\) are the direction cosines. For the given numbers, \[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{3}{4}\right)^2 = \frac{1}{2} + \frac{1}{3} + \frac{9}{16} = 0.5 + 0.3333 + 0.5625 = 1.3958 \neq 1. \] Thus, they do not satisfy the condition for direction cosines. ### (ii) The points (1, 2), (7, 6), and (4, 4) are collinear. False. To check if three points are collinear, we can compute the area of the triangle formed by them. If the area is zero, the points are collinear. Using the determinant method: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - ________ ______ _________ ________ _______ _____ ________ ____ ____.
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निर्देशांक अक्षों की दिशाओं को परिवर्तित किए बिना मूलबिंदु को (1,2,0) पर स्थानांतरित करके समीकरण को रूपांतरित कीजिए। यह नया समीकरण क्या निरुपित करता है? इसका स्थूल आरेख बनाइए।
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।
Identify and trace the conicoid y² + 3z2 = x. Describe its sections by the planes y = 0 and z = 0.
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.
Trace the conic x² 2xy + y² 3x + 2y + 3 = 0.
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.
बिंदु (2,-4,13) पर शांकवज के स्पर्श तल का समीकरण ज्ञात कीजिए। स्पर्श तल को ज्यामितीय रूप से दर्शाइए।