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