JEE MATH APEX

3D Geometry-JEE-Mains-PYQ’s

Mathematics Questions

Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line \frac{x-1}{1} = \frac{y}{2} = \frac{z-1}{1}. Then the distance of Q from the line \frac{x-9}{3} = \frac{y-9}{2} = \frac{z-5}{-2} is
(A) 8
(B) 7
(C) 6
(D) 5
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If the distances of the point (1, 2, a) from the line \frac{x-1}{1} = \frac{y}{2} = \frac{z-1}{1} along the lines L_1 : \frac{x-1}{3} = \frac{y-2}{4} = \frac{z-a}{b} and L_2 : \frac{x-1}{1} = \frac{y-2}{4} = \frac{z-a}{c} are equal, then a+b+c is equal to
(A) 4
(B) 6
(C) 7
(D) 5
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The sum of all values of \alpha, for which the shortest distance between the lines \frac{x+1}{\alpha} = \frac{y-2}{-1} = \frac{z-4}{-\alpha} and \frac{x}{\alpha} = \frac{y-1}{2} = \frac{z-1}{2\alpha} is \sqrt{2}, is
(A) -6
(B) -8
(C) 8
(D) 6
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Let the direction cosines of two lines satisfy the equations : 4l + m - n = 0 and 2mn + 10nl + 3lm = 0. Then the cosine of the acute angle between these lines is :
(A) \frac{10}{7\sqrt{38}}
(B) \frac{10}{\sqrt{38}}
(C) \frac{10}{3\sqrt{38}}
(D) \frac{20}{3\sqrt{38}}
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The vertices B and C of a triangle ABC lie on the line \frac{x}{1} = \frac{1-y}{-2} = \frac{z-2}{3}. The coordinates of A and B are (1, 6, 3) and (4, 9, \alpha) respectively and C is at a distance of 10 units from B. The area (in sq. units) of \triangle ABC is :
(A) 20\sqrt{13}
(B) 5\sqrt{13}
(C) 15\sqrt{13}
(D) 10\sqrt{13}
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Let L be the line \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} and let S be the set of all points (a, b, c) on L , whose distance from the line \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} along the line L is 7 . Then \sum_{(a,b,c)\in S} (a + b + c) is equal to :
(A) 28
(B) 6
(C) 40
(D) 34
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Let P(\alpha, \beta, \gamma) be the point on the line \frac{x-1}{2} = \frac{y+1}{-3} = z at a distance 4\sqrt{14} from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines \frac{x-\alpha}{1} = \frac{y-\beta}{2} = \frac{z-\gamma}{3} and \frac{x+5}{2} = \frac{y-10}{1} = \frac{z-3}{1}, is equal to
(A) 4\sqrt{\frac{7}{5}}
(B) 7\sqrt{\frac{5}{4}}
(C) 4\sqrt{\frac{5}{7}}
(D) 2\sqrt{\frac{7}{4}}
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If the image of the point \mathrm{P}(1, 2, a) in the line \frac{x-6}{3} = \frac{y-7}{2} = \frac{7-z}{2} is \mathrm{Q}(5, b, c), then a^2 + b^2 + c^2 is equal to
(A) 298
(B) 264
(C) 293
(D) 283
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Let the line L pass through the point (-3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point (-2, r, 1) is \sqrt{\frac{14}{3}}, then the sum of all possible values of r is :
(A) 16
(B) 12
(C) 6
(D) 10
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Let the line L_1 be parallel to the vector -3\hat{i} + 2\hat{j} + 4\hat{k} and pass through the point (2, 6, 7), and the line L_2 be parallel to the vector 2\hat{i} + \hat{j} + 3\hat{k} and pass through the point (4, 3, 5). If the line L_3 is parallel to the vector -3\hat{i} + 5\hat{j} + 16\hat{k} and intersects the lines L_1 and L_2 at the points C and D, respectively, then \left|\vec{CD}\right|^2 is equal to:
(A) 290
(B) 171
(C) 89
(D) 312
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