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NAVNEET 21 M. L. Q. SETS : MATHEMATICS & STATISTICS-STD. XII (Science), 3 The equation of the line passing through the point (x, y, 2) and having, SET, (Marks with option : 10), LINE, Remember:, C) 4, x-Xy-Y1, 3, direction ratios a, b, c is, xーX_, 1 1_2 – Z -i(sav), 4 The coordinates of any point on the line, 7., s The vector equation of the line in parametric form passing through the, points A (a) and B (b) is r= a + i(b- a)., 6 The non-parametric vector equation of the line passing through the points, 60° or 131, %3D, A(@) and B(b) is (r-a) x (5-a)-0., 1. The cquation of the line passing through the points A (x,, y, 2,) and, 8, -1, y-y, |, %3D, B (x, Y2, Z2) is, こュ-2」, 8. The length of the perpendicular from the point P(a,b,c) to the line, y-Yi -21 is given by, 1., Vlla - x,)? + (b - y,)²+(c-2,)*1-[(a -x,)l+(b-y,)m+(c-2)n], where I, m, n are the direction cosines of the line., 9. The length of perpendicular from the point P (z) to the line r=a + ìb is, given by, 10. The shortest distance between the lines r a, +2b, and r a,+ ub, is, |(a, -a,)·(b, x b,)|, given by d=, %3D, 166, Notes, 11. The shortest distance between the lines, メーズュ_ソーY2_ is given by, 1. Th, xーズ_yーソ」_ミー, and, %3D, %3D, 12, %3D, m2, n2, m1, 2. Si, 12, m2, n2, d%3D, (m,n2-m,n)²+(1,n -1,n2) +(1,m, -1,m,)?, |, 3. L, 12. The distance between the parallel lines r=a1+ ib and r a2+ub is, given by, (a, -a,) x b, d% =, %3D, 13. Lines r=a,+ ib, and r=a, + 2b, intersect each other if and only if, (az-a,)-(b, x b,) =0., 14. The lines, xーズ」_ソーY」 ミー, x-X2 y-Y2 -22 intersect, and, m1, m2, x2-X1 y2-y1 2-Z1, n2, each other if and only if, 14, m1, n1, =0., %3D, 12, m2, n2, Theory Question 3 marks, Q. Prove that the vector