Class 11 Mathematics Straight Lines MCQs Set 26

QUADRILATERALS AND AREA OF THE QUADRILATERALS

Question. The area enclosed by \( 2|x| + 3|y| \leq 6 \) is
(a) 3 sq. units
(b) 4 sq. units
(c) 12 sq. units
(d) 24 sq. units
Answer: (c) 12 sq. units

Question. The point on the line \( 3x - 2y = 1 \) which is closest to the origin is
(a) \( \left(\frac{3}{13}, \frac{-2}{13}\right) \)
(b) \( \left(\frac{5}{11}, \frac{2}{11}\right) \)
(c) \( \left(\frac{3}{5}, \frac{2}{5}\right) \)
(d) \( \left(\frac{3}{13}, \frac{2}{13}\right) \)
Answer: (a) \( \left(\frac{3}{13}, \frac{-2}{13}\right) \)

Question. The reflection of \( y = \sqrt{x} \) w.r.t. y-axis is
(a) \( y = -\sqrt{x} \)
(b) \( y = \sqrt{-x} \)
(c) \( y = -\sqrt{-x} \)
(d) \( x = \sqrt{y} \)
Answer: (b) \( y = \sqrt{-x} \)

Question. The points (-1, 1) and (1, -1) are symmetrical about the line
(a) \( y + x = 0 \)
(b) \( y = x \)
(c) \( x + y = 1 \)
(d) \( x - y = 1 \)
Answer: (b) \( y = x \)

Question. The equation of perpendicular bisectors of sides AB, BC of \( \Delta ABC \) are \( x - y - 5 = 0 \), \( x + 2y = 0 \) respectively and \( A(1, -2) \) then coordinate of C are
(a) (1, 0)
(b) (0, 1)
(c) (5, 0)
(d) (0, 0)
Answer: (c) (5, 0)

CENTROID, CIRCUMCENTRE, ORTHOCENTRE AND INCENTRE

Question. If one vertex of an equilateral triangle is the origin and side opposite to it has the equation \( x + y = 1 \), then the orthocentre of the triangle is
(a) \( \left(\frac{1}{3}, \frac{1}{3}\right) \)
(b) \( \left(\frac{2}{3}, \frac{2}{3}\right) \)
(c) (1, 1)
(d) (1, 3)
Answer: (a) \( \left(\frac{1}{3}, \frac{1}{3}\right) \)

Question. If the circum centre of the triangle lies at (0,0) and centroid is middle point of \( (a^2 + 1, a^2 + 1) \) and (2a, -2a) then the orthocentre lies on
(a) \( (a - 1)^2 x - (a + 1)^2 y = 0 \)
(b) \( (a - 1)^2 x + (a + 1)^2 y = 0 \)
(c) \( (a - 1)^2 x + (a + 1)^2 y + 56 = 0 \)
(d) \( (a - 1)^2 x + (a + 1)^2 y - 56 = 0 \)
Answer: (a) \( (a - 1)^2 x - (a + 1)^2 y = 0 \)

Question. The orthocentre of the triangle formed by the lines \( x + y = 1 \), \( 2x + 3y = 6 \) and \( 4x - y + 9 = 0 \) lies in quadrant number
(a) 1st
(b) IInd
(c) IIIrd
(d) IVth
Answer: (b) IInd

Question. If the straight lines \( 2x + 3y - 1 = 0 \), \( x + 2y - 1 = 0 \) and \( ax + by - 1 = 0 \) form a triangle with origin as orthocentre, then (a, b) is given by
(a) ( 6,4 )
(b) (-3,3 )
(c) ( -8,8 )
(d) ( 0,7 )
Answer: (c) ( -8,8 )

Question. In \( \Delta ABC \), equation to AB is \( 2x + 3y - 5 = 0 \), altitude through A is \( x - y + 4 = 0 \) and altitude through B is \( 2x - y - 1 = 0 \). Then the vertex C is
(a) \( \left(-\frac{1}{5}, \frac{9}{5}\right) \)
(b) \( \left(\frac{1}{5}, \frac{9}{5}\right) \)
(c) \( \left(\frac{1}{5}, -\frac{9}{5}\right) \)
(d) \( \left(-\frac{1}{5}, -\frac{9}{5}\right) \)
Answer: (b) \( \left(\frac{1}{5}, \frac{9}{5}\right) \)

Question. Centroid of the triangle, formed by the lines \( x + 2y - 5 = 0 \), \( 2x + y - 7 = 0 \), \( x - y + 1 = 0 \) is
(a) (1, 3)
(b) (3, 5)
(c) (2, 2)
(d) (1, 1)
Answer: (c) (2, 2)

ANGULAR BISECTORS

Question. The acute angle bisector between the lines \( 3x - 4y - 5 = 0 \), \( 5x + 12y - 26 = 0 \) is
(a) \( 7x - 56y + 32 = 0 \)
(b) \( 9x - 3y + 13 = 0 \)
(c) \( 14x - 112y + 65 = 0 \)
(d) \( 7x - 13y + 9 = 0 \)
Answer: (c) \( 14x - 112y + 65 = 0 \)

Question. The equation of the bisector of the angle between the lines \( x - 7y + 5 = 0 \), \( 5x + 5y - 3 = 0 \) which is the supplement of the angle containing the origin will be
(a) \( x + 3y - 2 = 0 \)
(b) \( x - 3y + 2 = 0 \)
(c) \( 3x - y + 1 = 0 \)
(d) \( 3x + y + 2 = 0 \)
Answer: (a) \( x + 3y - 2 = 0 \)

Question. Reflection of \( 3x + 4y + 5 = 0 \) w.r.to the line \( 2x + y + 1 = 0 \) is
(a) \( 2x + 1 = 0 \)
(b) \( 2x - 1 = 0 \)
(c) \( 5x - 1 = 0 \)
(d) \( 5x + 1 = 0 \)
Answer: (c) \( 5x - 1 = 0 \)

Question. Two sides of a Rhombus ABCD are parallel to the lines \( x - y = 5 \) and \( 7x - y = 3 \). The diagonals intersect at (2,1) then the equations of the diagonals are
(a) \( x - y = 1, 7x - y = 13 \)
(b) \( x + y = 3, x + 7y = 9 \)
(c) \( x + 2y = 4, 2x - y = 3 \)
(d) \( 3x + 4y = 10, 4x - 3y = 5 \)
Answer: (c) \( x + 2y = 4, 2x - y = 3 \)

Question. Let \( P = (-1, 0) \), \( Q = (0, 0) \) and \( R = (3, 3\sqrt{3}) \) be three points. Then the equation of the bisector of angle PQR is
(a) \( \frac{\sqrt{3}}{2}x + y = 0 \)
(b) \( x + \sqrt{3}y = 0 \)
(c) \( \sqrt{3}x + y = 0 \)
(d) \( x + \frac{\sqrt{3}}{2}y = 0 \)
Answer: (c) \( \sqrt{3}x + y = 0 \)

OPTIMIZATION AND REFLECTION IN SURFACE

Question. A ray of light along \( x + \sqrt{3}y = \sqrt{3} \) gets reflected upon reaching x-axis, the equation of the reflected ray is
(a) \( y = x + \sqrt{3} \)
(b) \( \sqrt{3}y = x - \sqrt{3} \)
(c) \( y = 3x - \sqrt{3} \)
(d) \( \sqrt{3}y = x - 1 \)
Answer: (b) \( \sqrt{3}y = x - \sqrt{3} \)

Question. Consider the points A(0,1) and B(2,0) and P be a point on the line \( 4x + 3y + 9 = 0 \). Coordinates of P such that \( |PA - PB| \) is maximum are
(a) \( \left(-\frac{24}{5}, \frac{17}{5}\right) \)
(b) \( \left(-\frac{84}{5}, \frac{13}{5}\right) \)
(c) \( \left(-\frac{6}{5}, \frac{17}{5}\right) \)
(d) (0, –3)
Answer: (a) \( \left(-\frac{24}{5}, \frac{17}{5}\right) \)

MISCELLANEOUS PROBLEMS

Question. A straight line which make equal intercepts on +ve x and y axes and which is at a distance '1' unit from the origin intersects the straight line \( y = 2x + 3\sqrt{2} \) at \( (x_0, y_0) \) then \( 2x_0 + y_0 = \)
(a) \( 3 + \sqrt{2} \)
(b) \( 2\sqrt{2} - 1 \)
(c) 1
(d) 0
Answer: (b) \( 2\sqrt{2} - 1 \)

Question. p is the length of the perpendicular drawn from the origin upon a straight line then the locus of mid point of the portion of the line intercepted between the coordinate axes is
(a) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{p^2} \)
(b) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{2}{p^2} \)
(c) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2} \)
(d) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{p} \)
Answer: (c) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2} \)

Question. Equation of the line passing through the point (2,3) and making intercept 2 units between the lines \( y + 2x = 3 \), \( y + 2x = 5 \) is
(a) \( x = 2 \)
(b) \( y = 3 \)
(c) \( x + y = 5 \)
(d) \( x + y = 7 \)
Answer: (a) \( x = 2 \)

Question. The number of lines that can be drawn through the point (4,-5) at a distance of 10 units from the point (1,3) is
(a) 0
(b) 1
(c) 2
(d) Infinite
Answer: (a) 0

Question. The number of circles that touch all the 3 lines \( 2x + y = 3 \), \( 4x - y = 3 \), \( x + y = 2 \) is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (b) 1

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

FOOT AND IMAGE

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 \( AB \) and \( AC \) of a triangle \( ABC \) are \( x-y-5=0 \) and \( x+2y=0 \) respectively. If \( A=(1,-2) \) then the equation of \( BC \) 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 midpoint 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) \( 2y=x-14 \)