Class 11 Mathematics Straight Lines MCQs Set 16

concurrency of Lines :

Question. The line parallel to the x-axis and passing through the intersection of the lines \( ax + 2by + 3b = 0 \) and \( bx - 2ay - 3a = 0 \), where \( (a, b) \neq (0, 0) \) is
(a) Above the x-axis at a distance of 3/2 from it
(b) Above the x-axis at a distance of 2/3 from it
(c) Below the x-axis at a distance of 3/2 from it
(d) Below the x-axis at a distance of 2/3 from it
Answer: (c) Below the x-axis at a distance of 3/2 from it

Question. If a, b, c form a G P with common ratio r, the sum of the ordinates of the points of intersection of the line \( ax + by + c = 0 \) and the curve \( x + 2y^2 = 0 \) is
(a) \( \frac{-r}{2} \)
(b) \( \frac{-r^2}{2} \)
(c) \( \frac{r}{2} \)
(d) \( \frac{r^2}{2} \)
Answer: (c) \( \frac{r}{2} \)

Question. Consider a family of straight lines \( (x + y) + \lambda(2x - y + 1) = 0 \). Find the equation of the straight line belonging to this family that is farthest from \( (1, -3) \).
(a) \( 3x - 3y + 2 = 0 \)
(b) \( 6x + 15y - 7 = 0 \)
(c) \( 5x + 2y + 1 = 0 \)
(d) \( 6x - 15y + 7 = 0 \)
Answer: (d) \( 6x - 15y + 7 = 0 \)

Question. If the line \( x = a + m, y = -2 \) and \( y = mx \) are concurrent, then least value of \( |a| \) is
(a) 0
(b) \( \sqrt{2} \)
(c) \( 2\sqrt{2} \)
(d) 2
Answer: (b) \( \sqrt{2} \)

Question. If \( a \neq b \neq c \), if \( ax + by + c = 0, bx + cy + a = 0 \) and \( cx + ay + b = 0 \) are concurrent. Then the value of \( 2^{a^2b^{-1}c^{-1} + b^2c^{-1}a^{-1} + c^2a^{-1}b^{-1}} \)
(a) 1
(b) 4
(c) 8
(d) 16
Answer: (c) 8

Question. Line \( ax + by + p = 0 \) makes angle \( \pi / 4 \) with \( x \cos \alpha + y \sin \alpha = p, p \in R^+ \). If these lines and the line \( x \sin \alpha - y \cos \alpha = 0 \) are concurrent, then
(a) \( a^2 + b^2 = 1 \)
(b) \( a^2 + b^2 = 2 \)
(c) \( 2(a^2 + b^2) = 1 \)
(d) \( a^2 - b^2 = 1 \)
Answer: (b) \( a^2 + b^2 = 2 \)

Angle between lines:

Question. If p, q, r are distinct, then \( (q - r)x + (r - p)y + (p - q) = 0 \) and \( (q^3 - r^3)x + (r^3 - p^3)y + (p^3 - q^3) = 0 \) represents the same line if
(a) \( p + q + r = 0 \)
(b) \( p = q = r \)
(c) \( p^2 + q^2 + r^2 = 0 \)
(d) \( p^3 + q^3 + r^3 = 0 \)
Answer: (a) \( p + q + r = 0 \)

Question. The lines \( (a + b)x + (a - b)y - 2ab = 0, (a - b)x + (a + b)y - 2ab = 0 \) and \( x + y = 0 \) form an isosceles triangle whose vertical angle is
(a) \( \frac{\pi}{2} \)
(b) \( \tan^{-1} \left( \frac{2ab}{a^2 - b^2} \right) \)
(c) \( \tan^{-1} \left( \frac{a}{b} \right) \)
(d) \( 2 \tan^{-1} \left( \frac{a}{b} \right) \)
Answer: (b) \( \tan^{-1} \left( \frac{2ab}{a^2 - b^2} \right) \)

Question. If \( 2(\sin a + \sin b)x - 2 \sin (a - b)y = 3 \) and \( 2(\cos a + \cos b)x + 2 \cos (a - b)y = 5 \) are perpendicular then \( \sin 2a + \sin 2b = \)
(a) \( \sin (a - b) - 2 \sin (a + b) \)
(b) \( \sin 2(a - b) - 2 \sin (a + b) \)
(c) \( 2 \sin (a - b) - \sin (a + b) \)
(d) \( \sin 2(a - b) - \sin (a + b) \)
Answer: (b) \( \sin 2(a - b) - 2 \sin (a + b) \)

Question. Two equal sides of an isosceles triangle are given by \( 7x - y + 3 = 0 \) and \( x + y - 3 = 0 \) and the third side passes through the point \( (1, 10) \) then the slope m of the third side is given by
(a) \( 3m^2 - 1 = 0 \)
(b) \( m^2 + 1 = 0 \)
(c) \( 3m^2 + 8m - 3 = 0 \)
(d) \( m^2 - 3 = 0 \)
Answer: (c) \( 3m^2 + 8m - 3 = 0 \)

Question. The diagonal of a square is \( 8x - 15y = 0 \) and one vertex of the square is \( (1, 2) \). Then the equations to the sides of the square passing through the vertex are
(a) \( 22x + 8y = 9, 22x - 8y = 52 \)
(b) \( 23x + 7y = 9, 7x - 23y = 52 \)
(c) \( 23x - 7y = 9, 7x + 23y = 53 \)
(d) \( 22x - 8y = 9, 22x + 8y = 52 \)
Answer: (c) \( 23x - 7y = 9, 7x + 23y = 53 \)

Triangles and area of the triangle:

Question. Area of triangle formed by angle bisectors of coordinate axes and the line \( x = 6 \) in sq.units is
(a) 36
(b) 18
(c) 72
(d) 9
Answer: (a) 36

Question. The quadratic equation whose roots are the x and y intercepts of the line passing through (1,1) and making a triangle of area A with the co -ordinate axes is
(a) \( x^2 + Ax + 2A = 0 \)
(b) \( x^2 - 2Ax + 2A = 0 \)
(c) \( x^2 - Ax + 2A = 0 \)
(d) \( (x - A)(x + A) = 0 \)
Answer: (b) \( x^2 - 2Ax + 2A = 0 \)

Question. A line passing through (3,4) meets the axes \( \vec{OX} \) and \( \vec{OY} \) at A and B respectively. The minimum area of the triangle OAB in square units is
(a) 8
(b) 16
(c) 24
(d) 32
Answer: (c) 24

Quadrilaterals and area of the quadrilaterals:

Question. The figure formed by the straight lines \( \sqrt{3}x + y = 0, \sqrt{3}y + x = 0, \sqrt{3}x + y = 1, \sqrt{3}y + x = 1 \) is
(a) a rectangle
(b) a square
(c) a rhombus
(d) parallelogram
Answer: (c) a rhombus

Question. Let the base of a triangle lie along the line \( x = a \) and be of length a. The area of this triangle is \( a^2 \), if the vertex lies on the line
(a) \( x + a = 0 \)
(b) \( x = 0 \)
(c) \( 2x - a = 0 \)
(d) \( x - a = 0 \)
Answer: (a) \( x + a = 0 \)

Question. The area bounded by \( y = |x| - 1, y = -|x| + 1 \)
(a) 1
(b) 2
(c) \( 2\sqrt{2} \)
(d) 4
Answer: (b) 2

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: (d) \( \left( \frac{3}{13}, \frac{-2}{13} \right) \)

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) \( (a-1)^2x - (a+1)^2y = 0 \)
(b) \( (a-1)^2x + (a+1)^2y = 0 \)
(c) \( (a-1)^2x + (a+1)^2y + 56 = 0 \)
(d) \( (a-1)^2x + (a+1)^2y - 56 = 0 \)
Answer: (a) \( (a-1)^2x - (a+1)^2y = 0 \)

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