JEE Mathematics Straight Lines MCQs Set D

Type – 1

Choose the most appropriate option (a, b, c or d).

Question. The sum of the intercepts made by the plane \( ax + by + cz = d \) on the three axes of reference is
(a) \( a + b + c \)
(b) \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \)
(c) \( d \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \)
(d) \( \frac{1}{d} \sqrt{a^2 + b^2 + c^2} \)
Answer: (c) \( d \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \)

Question. If the sum of the reciprocals of the intercepts made by the plane \( ax + by + cz = 1 \) on the three axes is 1 then the plane always passes through the point
(a) \( (2,-1,0) \)
(b) \( (1,1,1) \)
(c) \( (-1,-1,-1) \)
(d) \( \left( \frac{1}{2}, -1, \frac{1}{2} \right) \)
Answer: (b) \( (1,1,1) \)

Question. The direction cosines of the perpendicular from the origin to the plane \( 3x - y + 4z = 5 \) are
(a) \( 4, -1, 3 \)
(b) \( 3, -1, 4 \)
(c) \( \frac{3}{\sqrt{26}}, \frac{-1}{\sqrt{26}}, \frac{4}{\sqrt{26}} \)
(d) \( \frac{4}{\sqrt{26}}, \frac{-1}{\sqrt{26}}, \frac{3}{\sqrt{26}} \)
Answer: (c) \( \frac{3}{\sqrt{26}}, \frac{-1}{\sqrt{26}}, \frac{4}{\sqrt{26}} \)

Question. The length of the perpendicular from the origin to the plane \( 2x + 3y + \lambda z = 1 (\lambda > 0) \) is \( \frac{1}{5} \). Then \( \lambda \) is
(a) \( 2\sqrt{3} \)
(b) \( 3\sqrt{2} \)
(c) 0
(d) 1
Answer: (a) \( 2\sqrt{3} \)

Question. The direction cosines of the normal to the plane \( 5(x - 2) - 3(y - z) = 0 \) are
(a) \( 5, -3, 3 \)
(b) \( \frac{5}{\sqrt{43}}, \frac{-3}{\sqrt{43}}, \frac{3}{\sqrt{43}} \)
(c) \( \frac{1}{2}, \frac{-3}{10}, \frac{3}{10} \)
(d) \( 1, \frac{-3}{5}, \frac{3}{5} \)
Answer: (b) \( \frac{5}{\sqrt{43}}, \frac{-3}{\sqrt{43}}, \frac{3}{\sqrt{43}} \)

Question. A plane passing through the line joining the points \( A(1, -3, 5) \) and \( B(4, 1, -1) \) is turned about \( AB \) till it passes through the origin. The equation of the plane in the new position is
(a) \( 3x + 4y - 6z = 0 \)
(b) \( 2x - 21y + 13z = 0 \)
(c) \( 2x - 21y - 13z = 0 \)
(d) None of the options
Answer: (c) \( 2x - 21y - 13z = 0 \)

Question. The equations of a line passing through the point \( (-1, 0, 3) \) and perpendicular to the plane \( 4x + 3y - 5z = 12 \) are
(a) \( \frac{x-1}{4} = \frac{y}{3} = \frac{z+3}{-5} \)
(b) \( \frac{x+1}{4} = \frac{y}{3} = \frac{z-3}{-5} \)
(c) \( \frac{x+1}{-5} = \frac{y}{3} = \frac{z-3}{4} \)
(d) None of the options
Answer: (b) \( \frac{x+1}{4} = \frac{y}{3} = \frac{z-3}{-5} \)

Question. The equation of the plane passing through the line \( \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3} \) and parallel to the direction whose direction numbers are \( 3, 4, 2 \) is
(a) \( 14x - 5y - 11z = 19 \)
(b) \( 3x + 4y + 2z + 1 = 0 \)
(c) \( 2x - y + 3z = 3 \)
(d) None of the options
Answer: (a) \( 14x - 5y - 11z = 19 \)

Question. If the line \( \frac{x-1}{1} = \frac{y+1}{-2} = \frac{z+1}{\lambda} \) lies in the plane \( 3x - 2y + 5z = 0 \) then \( \lambda \) is
(a) 1
(b) \( -\frac{7}{5} \)
(c) \( \frac{5}{7} \)
(d) No possible value
Answer: (b) \( -\frac{7}{5} \)

Question. The equations of the line of intersection of the planes \( x + y + z = 2 \) and \( 3x - y + 2z = 5 \) in symmetric form are
(a) \( \frac{x - 7/4}{4} = \frac{y - 1/4}{-1} = \frac{z}{-3} \)
(b) \( \frac{x}{3} = \frac{y + 1/3}{1} = \frac{z - 7/3}{-4} \)
(c) \( \frac{x}{1} = \frac{3y + 1}{1} = \frac{3z - 7}{-4} \)
(d) None of the options
Answer: (b) \( \frac{x}{3} = \frac{y + 1/3}{1} = \frac{z - 7/3}{-4} \)

Question. The direction cosines of a line parallel to the planes \( 3x + 4y + z = 0 \) and \( x - 2y - 3z = 5 \) are
(a) \( (-1, 1, -1) \)
(b) \( \left( -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)
(c) \( \left( \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)
(d) No line possible
Answer: (c) \( \left( \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)

Question. If \( (3, \lambda, \mu) \) is a point on the line \( 2x + y + z - 3 = 0 = x - 2y + z - 1 \) then
(a) \( \lambda = -\frac{8}{3}, \mu = -\frac{1}{3} \)
(b) \( \lambda = \frac{1}{3}, \mu = -\frac{8}{3} \)
(c) \( \lambda = -1, \mu = -5 \)
(d) \( \lambda = -5, \mu = -1 \)
Answer: (b) \( \lambda = \frac{1}{3}, \mu = -\frac{8}{3} \)

Question. The equations of the perpendicular from the point \( (-2, 4, 1) \) to the plane \( 7x - 2y + 3z = 1 \) are
(a) \( \frac{x-5}{7} = \frac{y-2}{-2} = \frac{z-4}{3} \)
(b) \( \frac{x-2}{7} = \frac{y+4}{-2} = \frac{z+1}{3} \)
(c) \( \frac{x+2}{1} = \frac{y-4}{4} = \frac{z-1}{-2} \)
(d) None of the options
Answer: (a) \( \frac{x-5}{7} = \frac{y-2}{-2} = \frac{z-4}{3} \)

Question. \( P \) is a point on the y-z plane, making equal angles with the y-axis and z-axis and at a distance 2 from the origin. \( M \) is the foot of the perpendicular from \( P \) to the plane \( 3x + y - z\sqrt{2} = 2\sqrt{2} \). The coordinates of \( M \) are
(a) \( \left( 1, \frac{5}{3}, \frac{\sqrt{2}}{3} \right) \)
(b) \( (1, -3, -2) \)
(c) \( \left( \frac{1}{\sqrt{2}}, \frac{5}{3\sqrt{2}}, \frac{1}{3\sqrt{2}} \right) \)
(d) None of the options
Answer: (d) None of the options

Question. The distance of the point \( (2, 0, -3) \) from the plane \( 5x - 12y = 0 \) is
(a) \( \frac{10}{13} \)
(b) \( \frac{46}{13} \)
(c) \( \frac{36}{13} \)
(d) None of the options
Answer: (a) \( \frac{10}{13} \)

Question. The image of the point \( P(\alpha, \beta, \gamma) \) by the plane \( lx + my + nz = 0 \) is the point \( Q(\alpha', \beta', \gamma') \). Then
(a) \( \alpha^2 + \beta^2 + \gamma^2 = l^2 + m^2 + n^2 \)
(b) \( \alpha^2 + \beta^2 + \gamma^2 = \alpha'^2 + \beta'^2 + \gamma'^2 \)
(c) \( \alpha\alpha' + \beta\beta' + \gamma\gamma' = 0 \)
(d) \( l(\alpha - \alpha') + m(\beta - \beta') + n(\gamma - \gamma') = 0 \)
Answer: (b) \( \alpha^2 + \beta^2 + \gamma^2 = \alpha'^2 + \beta'^2 + \gamma'^2 \)

Question. The image of the point \( (2, -1, 1) \) by the plane \( 3x + 4y - 5z = 0 \) is
(a) \( (-2, 1, -1) \)
(b) \( \left( \frac{2}{3}, \frac{-1}{4}, \frac{-1}{5} \right) \)
(c) \( \left( \frac{59}{25}, \frac{13}{25}, \frac{2}{5} \right) \)
(d) None of the options
Answer: (c) \( \left( \frac{59}{25}, \frac{13}{25}, \frac{2}{5} \right) \)

Question. If the image of the point \( (1,1,1) \) by a plane be \( (3, -1, 5) \) then the equation of the plane is
(a) \( x - y + 2z = 8 \)
(b) \( x - y + 2z = 16 \)
(c) \( x - y + 2z = 14 \)
(d) None of the options
Answer: (a) \( x - y + 2z = 8 \)

Question. The angle between the line \( x = y = z \) and the plane \( 4x - 3y + 5z = 2 \) is
(a) \( \cos^{-1} \frac{\sqrt{6}}{5} \)
(b) \( \sin^{-1} \frac{\sqrt{6}}{5} \)
(c) \( \frac{\pi}{2} \)
(d) \( \sin^{-1} \frac{1}{\sqrt{6}} \)
Answer: (b) \( \sin^{-1} \frac{\sqrt{6}}{5} \)

Question. The equation of the plane passing through the origin and containing the line of intersection of the planes \( 5x + y - 3z = 2 \) and \( x + 2y + 3z = 1 \) is
(a) \( 2x + y = 1 \)
(b) \( x - y - 3z = 0 \)
(c) \( 4x - y - 6z = 0 \)
(d) \( 7x + 5y + 3z = 0 \)
Answer: (b) \( x - y - 3z = 0 \)

Question. What is the equation of the plane passing through the line of intersection of the planes \( x - y + 3z = 4 \) and \( 2x + y + 3z = 5 \) and parallel to the plane \( x + y + z = 1 \)?
(a) \( x + y + z = 2 \)
(b) \( x + y + z + 2 = 0 \)
(c) \( 2x = y + z \)
(d) No plane exists
Answer: (a) \( x + y + z = 2 \)

Question. What is the equation of the plane passing through the line \( 3x + y - 5z = 2 = x - 2y + 3z \) and perpendicular to the plane \( x - y + z = 3 \)?
(a) \( 2x + 3y + z = 2 \)
(b) \( 3x + 2y - z = 2 \)
(c) \( 7(x - z) = 6 \)
(d) No plane exists
Answer: (c) \( 7(x - z) = 6 \)

Question. The equation of the plane passing through the line \( x + y - 2 = 0 = x - y - 2z \) and at a distance 1 from the point \( (0, 1, 1) \) is
(a) \( 2x + y - z - 3 = \sqrt{3}(2 - x - y) \)
(b) \( x - y - 2z + \sqrt{3}(x + y - 2) = 0 \)
(c) \( x + y - 2 = \sqrt{3}(x - y - 2z) \)
(d) None of the options
Answer: (a) \( 2x + y - z - 3 = \sqrt{3}(2 - x - y) \)

Question. The angle between the planes \( x + y + z = 0 \) and \( 3x - 4y + 5z = 0 \) is
(a) \( \cos^{-1} \left( \frac{1}{5} \sqrt{\frac{2}{3}} \right) \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{3} \)
(d) \( \cos^{-1} \frac{2}{5}\sqrt{\frac{2}{3}} \)
Answer: (d) \( \cos^{-1} \frac{2}{5}\sqrt{\frac{2}{3}} \)

Question. The variable plane \( (2k + 1)x + (3 - \lambda)y + z = 4 \) always passes through the line
(a) \( \frac{x}{0} = \frac{y}{0} = \frac{z - 4}{1} \)
(b) \( \frac{x}{1} = \frac{y}{2} = \frac{z}{-3} \)
(c) \( \frac{x}{1} = \frac{y}{2} = \frac{z - 4}{-7} \)
(d) None of the options
Answer: (c) \( \frac{x}{1} = \frac{y}{2} = \frac{z - 4}{-7} \)

Question. The distance between the planes \( 4x - 5y + 3z = 5 \) and \( 4x - 5y + 3z + 2 = 0 \) is
(a) \( \frac{7}{2\sqrt{5}} \)
(b) 7
(c) \( \frac{7}{5\sqrt{2}} \)
(d) 3
Answer: (c) \( \frac{7}{5\sqrt{2}} \)

Question. The distance between the planes \( x + 2y - 3z - 4 = 0 \) and \( 2x + 4y - 6z = 7 \) along the line \( \frac{x}{1} = \frac{y}{-3} = \frac{z}{2} \) is
(a) \( \frac{19}{22} \)
(b) \( \frac{3}{22} \)
(c) 5
(d) None of the options
Answer: (d) None of the options

Question. The shortest distance between the lines \( x-y = 0 = 2x + z \) and \( x + y - 2 = 0 = 3x - y + z - 1 \) is
(a) \( 11x - 3y = 0 \)
(b) \( 3x + 11y = 0 \)
(c) ...
Answer: (b) \( 3x + 11y = 0 \)

Question. Which of the following planes is equally inclined to the planes \( 4x + 3y - 5z = 0 \) and \( 5x - 12y + 13z = 0 \)?
(a) \( 11x - 3y = 0 \)
(b) \( 3x + 11y = 0 \)
(c) \( 3x + 11y = 65z \)
(d) None of the options
Answer: (a) \( 11x - 3y = 0 \)

Question. The equation of the plane bisecting the angle between the planes \( 3x + 4y = 4 \) and \( 6x - 2y + 3z + 5 = 0 \) that contains the origin, is
(a) \( 9x - 38y + 15z + 43 = 0 \)
(b) \( 51x + 18y + 15z = 13 \)
(c) \( 9x + 2y + 3z + 1 = 0 \)
(d) None of the options
Answer: (b) \( 51x + 18y + 15z = 13 \)

Question. The equation of the plane bisecting the acute angle between the planes \( x - y + z - 1 = 0 \) and \( x + y + z = 2 \) is
(a) \( x + z = \frac{3}{2} \)
(b) \( 2y = 1 \)
(c) \( x - y - z = 3 \)
(d) None of the options
Answer: (a) \( x + z = \frac{3}{2} \)

Question. The direction cosines of the projection of the line \( \frac{x}{-2} = \frac{y-1}{1} = \frac{z+1}{-1} \) on the plane \( 2x + y - 3z = 5 \) are
(a) \( 2, -1, 1 \)
(b) \( \frac{2}{7}, \frac{-1}{7}, \frac{1}{7} \)
(c) \( \frac{-2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}} \)
(d) \( \frac{2}{\sqrt{6}}, \frac{-1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \)
Answer: (d) \( \frac{2}{\sqrt{6}}, \frac{-1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \)

Question. Two systems of rectangular axes have the same origin. If a plane cuts them at distances \( a, b, c \) and \( a', b', c' \) from the origin then
(a) \( a^{-2} + b^{-2} - c^{-2} + a'^{-2} + b'^{-2} - c'^{-2} = 0 \)
(b) \( a^{-2} - b^{-2} - c^{-2} + a'^{-2} - b'^{-2} - c'^{-2} = 0 \)
(c) \( a^{-2} + b^{-2} + c^{-2} - a'^{-2} - b'^{-2} - c'^{-2} = 0 \)
(d) None of the options
Answer: (c) \( a^{-2} + b^{-2} + c^{-2} - a'^{-2} - b'^{-2} - c'^{-2} = 0 \)

Question. The lines \( x = ay + b, z = cy + d \) and \( x = a'y + b', z = c'y + d' \) will be perpendicular if and only if
(a) \( aa' + bb' + cc' = 0 \)
(b) \( (a + a')(b + b') + c + c' = 0 \)
(c) \( aa' + cc' + 1 = 0 \)
(d) \( aa' + bb' + cc' + 1 = 0 \)
Answer: (c) \( aa' + cc' + 1 = 0 \)

Question. Which of the following planes intersects the planes \( x - y + 2z = 3 \) and \( 4x + 3y - z = 1 \) along the same line?
(a) \( 11x + 10y - 5z = 0 \)
(b) \( 7x + 7y - 4z = 0 \)
(c) \( 5x + 2y + z = 2 \)
(d) None of the options
Answer: (a) \( 11x + 10y - 5z = 0 \)

Question. The line \( \frac{x-1}{2} = \frac{y}{-1} = \frac{z+2}{2} \) cuts the plane \( x + y + z = 1 \) at \( P \). If the foot of the perpendicular from \( P \) to a plane be \( (3, -4, 1) \) then the equation of the plane is
(a) \( 3x - 2y - 2 = 0 \)
(b) \( 2x - y + 2z = 12 \)
(c) \( 2x - 10y + 5z = 51 \)
(d) None of the options
Answer: (c) \( 2x - 10y + 5z = 51 \)

Question. A variable plane at a distance of 1 unit from the origin cuts the coordinate axes at \( A, B \) and \( C \). If the centroid \( D(x, y, z) \) satisfies the relation \( x^{-2} + y^{-2} + z^{-2} = k \) then the value of \( k \) is
(a) 3
(b) 1
(c) 6
(d) 9
Answer: (d) 9

Type 2

Choose the correct options. One or more options may be correct.

Question. A plane through the line \( \frac{x-1}{1} = \frac{y+1}{-2} = \frac{z}{1} \) has the equation
(a) \( x + y + z = 0 \)
(b) \( 3x + 2y - z = 1 \)
(c) \( 4x + y - 2z = 3 \)
(d) \( 3x + 2y + z = 0 \)
Answer: (a) \( x + y + z = 0 \) and (c) \( 4x + y - 2z = 3 \)

Question. The equation of a plane is \( 2x - y - 3z = 5 \) and \( A(1, 1, 1), B(2, 1, -3), C(1, -2, -2) \) and \( D(-3, 1, 2) \) are four points. Which of the following line segments are intersected by the plane?
(a) \( AD \)
(b) \( AB \)
(c) \( AC \)
(d) \( BC \)
Answer: (b) \( AB \) and (c) \( AC \)