Class 11 Mathematics Straight Lines MCQs Set 18

Quadrilaterals and area of the quadrilaterals:

Question. The area enclosed with in the curve \( |x| + |y| = 1 \) is
(a) 1
(b) 2
(c) \( 2\sqrt{2} \)
(d) 4
Answer: (b) 2

Question. Foot of the perpendicular of \( (6,8) \) in the line \( x = y \) is
(a) \( (6,6) \)
(b) \( (7,7) \)
(c) \( (-6, -6) \)
(d) \( (-7, -7) \)
Answer: (b) \( (7,7) \)

Question. P is the midpoint of the part of the line \( 3x + y - 2 = 0 \) intercepted between the axes. Then the image of P in origin is
(a) \( \left(-1, -\frac{1}{3}\right) \)
(b) \( \left(-\frac{1}{3}, -4\right) \)
(c) \( \left(-\frac{1}{3}, -1\right) \)
(d) \( (-2, -3) \)
Answer: (c) \( \left(-\frac{1}{3}, -1\right) \)

Question. The image of the point \( P(3,5) \) with respect to the line \( y = x \) is the point Q and the image of Q with respect to the line \( y = 0 \) is the point \( R(a, b) \), then \( (a, b) = \)
(a) \( (5,3) \)
(b) \( (5,-3) \)
(c) \( (-5,3) \)
(d) \( (-5,-3) \)
Answer: (b) \( (5,-3) \)

Question. The equation of perpendicular bisector of \( \overline{AB} \) and \( \overline{AC} \) of a triangle ABC are \( x - y - 5 = 0 \) and \( x + 2y = 0 \) respectively. If \( A = (1, -2) \) then the equation of \( \overline{BC} \) is
(a) \( 14x + 2y - 41 = 0 \)
(b) \( 11x + 2y - 25 = 0 \)
(c) \( 2x - y - 10 = 0 \)
(d) \( 14x - 23y + 40 = 0 \)
Answer: (b) \( 11x + 2y - 25 = 0 \)

Centroid, circumcentre, orthocentre and incentre:

Question. Let \( O(0,0), P(3,4), Q(6,0) \) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are
(a) \( \left(\frac{4}{3}, 3\right) \)
(b) \( \left(3, \frac{2}{3}\right) \)
(c) \( \left(3, \frac{4}{3}\right) \)
(d) \( \left(\frac{4}{3}, \frac{2}{3}\right) \)
Answer: (c) \( \left(3, \frac{4}{3}\right) \)

Question. If the circumcentre of the triangle lies at \( (0,0) \) and centroid is mid point of the line joining the points \( (2,3) \) and \( (4,7) \), then its orthocentre lies on the line
(a) \( 5x - 3y = 0 \)
(b) \( 5x - 3y + 6 = 0 \)
(c) \( 5x + 3y = 0 \)
(d) \( 5x + 3y + 6 = 0 \)
Answer: (a) \( 5x - 3y = 0 \)

Question. The orthocentre of the triangle formed by the lines \( x+y=6, 2x+y=4 \) and \( x+2y=5 \) is
(a) \( (10, -11) \)
(b) \( (-10, 11) \)
(c) \( (11, -10) \)
(d) \( (-11, 10) \)
Answer: (d) \( (-11, 10) \)

Question. The equation \( x + 2y = 3 \) represents the side BC of \( \Delta ABC \); where co-ordinates of A are \( (1,2) \). If the x-coordinate of the orthocentre of \( \Delta ABC \) is 3 then the y-coordinates of the orthocentre is:
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (b) 6

Question. The vertices A,B of a triangle are \( (2, 5), (4, -11) \). If C moves on the line \( L \equiv 9x + 7y + 4 = 0 \), then the locus of centroid of triangle ABC is parallel to
(a) AB
(b) AC
(c) BC
(d) L
Answer: (d) L

Question. Two sides of a triangle are \( y = m_1x \) and \( y = m_2x \); \( m_1, m_2 \) are the roots of the equation \( x^2 + ax – 1 = 0 \). For all values of ‘a’ the orthocentre of the triangle lies at
(a) \( (1, 1) \)
(b) \( (2, 2) \)
(c) \( \left(\frac{3}{2}, \frac{3}{2}\right) \)
(d) \( (0, 0) \)
Answer: (d) \( (0, 0) \)

Angular bisectors :

Question. Equation of the line equidistant from the lines \( 2x+y+4=0, 3x+6y-5=0 \) is
(a) \( 3x - 3y + 17 = 0 \)
(b) \( 5x + 7y - 5 = 0 \)
(c) \( 3x - 3y + 19 = 0 \)
(d) \( 9x - 9y + 17 = 0 \)
Answer: (a) \( 3x - 3y + 17 = 0 \)

Question. Find the equation of the bisector of the angle between the lines \( x+2y–11=0, 3x–6y–5=0 \) which contains the point \( (1,–3) \).
(a) \( 2x - 19 = 0 \)
(b) \( 2x + 19 = 0 \)
(c) \( 3x - 19 = 0 \)
(d) \( 3x + 19 = 0 \)
Answer: (c) \( 3x - 19 = 0 \)

Question. The line \( 3x-3y+17=0 \) bisects the angle between a pair of lines of which one line is \( 2x+y+4=0 \), then the equation to the other line is
(a) \( 3x + 6y - 5 = 0 \)
(b) \( 3x + 6y - 7 = 0 \)
(c) \( 7x + 14y = 0 \)
(d) \( 4x - y + 3 = 0 \)
Answer: (a) \( 3x + 6y - 5 = 0 \)

Question. The equation of a straight line passing through the point \( (4,5) \) and equally inclined to the lines \( 3x = 4y + 7 \) and \( 5y = 12x + 6 \) is
(a) \( 9x - 7y = 1 \)
(b) \( 9x + 7y = 1 \)
(c) \( 7x - 9y = 1 \)
(d) \( 7x - 9y = 17 \)
Answer: (a) \( 9x - 7y = 1 \)

Question. If \( 2x + y - 4 = 0 \) is bisector of the angle between the lines \( a(x–1) + b(y–2) = 0, c(x–1) + d(y–2) = 0 \), then the other bisector is
(a) \( x – 2y + 1 = 0 \)
(b) \( x – 2y – 3 = 0 \)
(c) \( x – 2y + 3 = 0 \)
(d) \( x - 2y – 5 = 0 \)
Answer: (c) \( x – 2y + 3 = 0 \)

Optimization and reflection in surface:

Question. A ray of light passing through the point \( (8,3) \) and is reflected at \( (14,0) \) on x axis. Then the equation of the reflected ray
(a) \( x + y = 14 \)
(b) \( x - y = 14 \)
(c) \( 2y = x - 14 \)
(d) \( 3y = x - 14 \)
Answer: (c) \( 2y = x - 14 \)

Question. Let \( P(1,1) \) and \( Q(3,2) \) be given points. The point R on the x-axis such that \( PR+RQ \) is minimum is
(a) \( \left(\frac{5}{3}, 0\right) \)
(b) \( (2, 0) \)
(c) \( (3, 0) \)
(d) \( \left(\frac{3}{2}, 0\right) \)
Answer: (a) \( \left(\frac{5}{3}, 0\right) \)

Miscellaneous problems:

Question. The vertices of triangle are \( A(m, n) \), \( B(12, 19) \) and \( C(23, 20) \), where m and n are integer. If its area is 70 and the slope of the median through A is \( –5 \), then \( m+n \) is
(a) 47
(b) 27
(c) 107
(d) 43
Answer: (c) 107

Question. Number of circles touching the lines \( 3x + 4y – 1 = 0 \), \( 4x – 5y + 2 = 0 \) and \( 6x + 8y + 3 = 0 \) is
(a) 0
(b) 2
(c) 4
(d) infinite
Answer: (b) 2

Question. The point on the line \( 3x + 4y = 5 \) which is equidistant from \( (1,2) \) and \( (3,4) \) is
(a) \( (7, -4) \)
(b) \( (15, -10) \)
(c) \( \left(\frac{1}{7}, \frac{8}{7}\right) \)
(d) \( \left(0, \frac{5}{4}\right) \)
Answer: (b) \( (15, -10) \)

Question. The number of points \( p(x,y) \) with natural numbers as coordinates that lie inside the quadrilateral formed by the lines \( 2x + y = 2, x = 0, y = 0 \) and \( x + y = 5 \) is
(a) 12
(b) 10
(c) 6
(d) 4
Answer: (c) 6

Question. A point moves in the xy plane such that the sum of its distance from two mutually perpendicular lines is always equal to 5 units. The area (in square units) enclosed by the locus of the point
(a) \( \frac{25}{4} \)
(b) 25
(c) 50
(d) 100
Answer: (c) 50

Question. The line \( 3x - 2y = 24 \) meets x-axis at A and y-axis at B. The perpendicular bisector of AB meets the line through (0, -1) and parallel to x-axis at C. Then C is
(a) \( \left( \frac{-7}{2}, -1 \right) \)
(b) \( \left( \frac{-15}{2}, -1 \right) \)
(c) \( \left( \frac{-11}{2}, -1 \right) \)
(d) \( \left( \frac{-13}{2}, -1 \right) \)
Answer: (a) \( \left( \frac{-7}{2}, -1 \right) \)

Question. A square of side " a " lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle \( \alpha \) where \( 0 < \alpha < \frac{\pi}{4} \) with the positive direction of x-axis. The equation of its diagonal not passing through the origin is
(a) \( y(\cos \alpha + \sin \alpha) + x(\cos \alpha - \sin \alpha) = a \)
(b) \( y(\cos \alpha - \sin \alpha) - x(\sin \alpha - \cos \alpha) = a \)
(c) \( y(\cos \alpha + \sin \alpha) + x(\sin \alpha - \cos \alpha) = a \)
(d) \( y(\cos \alpha + \sin \alpha) + x(\cos \alpha + \sin \alpha) = a \)
Answer: (a) \( y(\cos \alpha + \sin \alpha) + x(\cos \alpha - \sin \alpha) = a \)

Question. A particle is moving in a straight line and at some moment it occupied the positions (5,2) and (-1,-2). Then the position of the particle when it is on x-axis is
(a) (-2, 0)
(b) (0, 2)
(c) (2, 0)
(d) (4, 0)
Answer: (c) (2, 0)

Question. If PS is the median of the triangle with vertices P(2,2), Q(6,-1) and R(7,3) then the equation of the line passing through (1,-1) and parallel to PS
(a) \( 4x - 7y - 11 = 0 \)
(b) \( 2x + 9y + 7 = 0 \)
(c) \( 4x + 7y + 3 = 0 \)
(d) \( 2x - 9y - 11 = 0 \)
Answer: (b) \( 2x + 9y + 7 = 0 \)

Question. The Point P(2, 1) is shifted by \( 3\sqrt{2} \) parallel to the line \( x + y = 1 \), in the direction of increasing ordinate, to reach Q. The image of Q by the line \( x + y = 1 \) is
(a) \( (5, -2) \)
(b) \( (-1, 4) \)
(c) \( (3, -4) \)
(d) \( (-3, 2) \)
Answer: (d) \( (-3, 2) \)

Question. Distance of origin from line \( (1 + \sqrt{3})y + (1 - \sqrt{3})x = 10 \) along the line \( y = \sqrt{3}x + k \) is
(a) \( 5/\sqrt{2} \)
(b) \( 5/\sqrt{2} + k \)
(c) 10
(d) 5
Answer: (d) 5

Question. One of the diagonals of a square is the portion of the line \( \frac{x}{2} + \frac{y}{3} = 1 \) intercepted between the axes. Then the extremities of the other diagonal are
(a) \( (5,5), (-1,1) \)
(b) \( (0,0), (4,6) \)
(c) \( (0,0), (-1,1) \)
(d) \( (5,5), (4,6) \)
Answer: (a) \( (5,5), (-1,1) \)

Question. If the line \( y = \sqrt{3}x \) cuts the curve \( x^3 + y^3 + 3xy + 5x^2 + 3y^2 + 4x + 5y - 1 = 0 \) at the points A, B, C then OA.OB.OC is
(a) \( \frac{4}{13}(3\sqrt{3} - 1) \)
(b) \( 3\sqrt{3} + 1 \)
(c) \( \frac{2}{\sqrt{3}} + 7 \)
(d) \( 3\sqrt{3} - 1 \)
Answer: (a) \( \frac{4}{13}(3\sqrt{3} - 1) \)

Question. Each side of a square is of length 4. The centre of the square is (3, 7) and one of its diagonals is parallel to y=x. Then co-ordinates of its vertices are
(a) (1,5), (1,9), (5,9), (5,5)
(b) (2,5), (2,7), (4,7), (4,4)
(c) (2,5), (2,6), (3,5), (3,6)
(d) (5,2), (6,2), (5,3), (6,3)
Answer: (a) (1,5), (1,9), (5,9), (5,5)

Question. If the line \( y - \sqrt{3}x + 3 = 0 \) cuts the curve \( y^2 = x + 2 \) at A and B and point on the line P is \( (\sqrt{3}, 0) \) then \( |PA.PB| = \)
(a) \( \frac{4(\sqrt{3} + 2)}{3} \)
(b) \( \frac{4(2 - \sqrt{3})}{3} \)
(c) \( \frac{4\sqrt{3}}{2} \)
(d) \( \frac{2(\sqrt{3} + 2)}{3} \)
Answer: (c) \( \frac{4\sqrt{3}}{2} \)

Question. Points A and B are in the first quadrant; point 'O' is the origin. If the slope of OA is 1, slope of OB is 7 and OA = OB, then the slope of AB is
(a) \( -1/5 \)
(b) \( -1/4 \)
(c) \( -1/3 \)
(d) \( -1/2 \)
Answer: (d) \( -1/2 \)

Question. The line joining the points A(3,0) and B(5,2) is rotated about A in the anticlockwise direction through an angle of \( 15^\circ \). If B goes to C in the new position now the line joining A and C is rotated about A in the anticlockwise direction through an angle of \( 45^\circ \) of C goes to D in the new position, then the coordinates of D are
(a) \( (4 - \sqrt{3}, \sqrt{3} - 1) \)
(b) \( (4 + \sqrt{3}, \sqrt{3} - 1) \)
(c) \( (4 - \sqrt{3}, \sqrt{3} + 1) \)
(d) \( (4 + \sqrt{3}, \sqrt{3} + 1) \)
Answer: (b) \( (4 + \sqrt{3}, \sqrt{3} - 1) \)

Question. The equation of the line passing through (1,2) and having a distance equal to 7 units from the point (8,9) is
(a) \( y = 3x - 1 \)
(b) \( y = 2 \)
(c) \( x = 1 \)
(d) \( x + y = 3 \)
Answer: (b) \( y = 2 \)

Question. Find the values of non-negative real numbers \( h_1, h_2, h_3, k_1, k_2, k_3 \) such that the algebraic sum of the perpendiculars drawn from points \( (2, k_1), (3, k_2), (7, k_3), (h_1, 4), (h_2, 5), (h_3, -3) \) on a variable line passing through \( (2, 1) \) is zero.
(a) \( h_1 = h_2 = h_3 = k_1 = k_2 = k_3 = 0 \)
(b) \( h_1 = h_2 = h_3 = k_1 = k_2 = k_3 = 1 \)
(c) \( h_1 = h_2 = h_3 = k_1 = k_2 = k_3 = 2 \)
(d) \( h_1 = h_2 = h_3 = k_1 = k_2 = k_3 = 4 \)
Answer: (a) \( h_1 = h_2 = h_3 = k_1 = k_2 = k_3 = 0 \)

Question. Three lines \( x + 2y + 3 = 0 \); \( x + 2y - 7 = 0 \) and \( 2x - y - 4 = 0 \) form the three sides of two squares. The equation to the fourth side of each square is
(a) \( 2x - y + 14 = 0 \) and \( 2x - y + 6 = 0 \)
(b) \( 2x - y + 14 = 0 \) and \( 2x - y - 6 = 0 \)
(c) \( 2x - y - 14 = 0 \) and \( 2x - y - 6 = 0 \)
(d) \( 2x - y - 14 = 0 \) and \( 2x - y + 6 = 0 \)
Answer: (d) \( 2x - y - 14 = 0 \) and \( 2x - y + 6 = 0 \)

Question. The line \( L \) given by \( \frac{x}{5} + \frac{y}{b} = 1 \) passes through the point \( (13, 32) \). The line \( K \) is parallel to \( L \) and has the equation \( \frac{x}{c} + \frac{y}{3} = 1 \). Then the distance between \( L \) and \( K \) is
(a) \( \sqrt{17} \)
(b) \( \frac{17}{\sqrt{15}} \)
(c) \( \frac{23}{\sqrt{17}} \)
(d) \( \frac{23}{\sqrt{15}} \)
Answer: (c) \( \frac{23}{\sqrt{17}} \)

Question. If the line \( 2x + y = k \) passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then \( k \) equals
(a) \( 29/5 \)
(b) 5
(c) 6
(d) \( 11/5 \)
Answer: (c) 6