Class 11 Mathematics Straight Lines MCQs Set 14

Question. The area of the triangle formed by the lines x=0; y=0 and \( x \sin 18^\circ + y \cos 36^\circ + 1 = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. If a straight line perpendicular to \( 3x-4y-6=0 \) forms a triangle with the coordinate axes whose area is 6 sq. units, then the equation of the straight line (s) is
(a) \( x-2y=6 \)
(b) \( 4x+3y=12 \)
(c) \( 4x+3y+24=0 \)
(d) \( 3x-4y=12 \)
Answer: (b) \( 4x+3y=12 \)

Question. The equation of base of an equilateral triangle is \( x+y=2 \) and the vertex is (2, -1). Then area of triangle is
(a) \( 2\sqrt{3} \)
(b) \( \sqrt{3}/6 \)
(c) \( 1\sqrt{3} \)
(d) \( 2\sqrt{3} \)
Answer: (b) \( \sqrt{3}/6 \)

Question. The quadrilateral formed by the lines \( 2x-5y+7=0 \), \( 5x+2y-1=0 \), \( 2x-5y+2=0 \), \( 5x+2y+3=0 \) is
(a) Rectangle
(b) Square
(c) Parallelogram
(d) Rhombus
Answer: (a) Rectangle

Question. The diagonals of a parallelogram PQRS are along the lines \( x + y = 3 \) and \( 6x - 2y = 7 \). Then PQRS must be :
(a) rectangle
(b) square
(c) cyclic quadrilateral
(d) rhombus
Answer: (d) rhombus

Question. Foot of the perpendicular of origin on the line joining the points \( (a \cos \theta, a \sin \theta) \), \( (a \cos \phi, a \sin \phi) \) is
(a) \( (\cos \theta + \cos \phi, \sin \theta + \sin \phi) \)
(b) \( (\cos \theta - \cos \phi, \sin \theta - \sin \phi) \)
(c) \( \left( \frac{a(\cos \theta + \cos \phi)}{2}, \frac{a(\sin \theta + \sin \phi)}{2} \right) \)
(d) \( (\cos \theta \cos \phi, \sin \theta \sin \phi) \)
Answer: (c) \( \left( \frac{a(\cos \theta + \cos \phi)}{2}, \frac{a(\sin \theta + \sin \phi)}{2} \right) \)

Question. Suppose A, B are two points on \( 2x-y+3=0 \) and P(1,2) is such that PA=PB. Then the mid point of AB is
(a) \( (-1/5, 13/5) \)
(b) \( (-7/5, 9/5) \)
(c) \( (7/5, -9/5) \)
(d) \( (-7/5, -9/5) \)
Answer: (a) \( (-1/5, 13/5) \)

Question. A line passing through the points (7,2), (-3,2) then the image of the line in x-axis is
(a) y = 4
(b) y = 9
(c) y = –1
(d) y = –2
Answer: (d) y = –2

Question. Image of the curve \( x^2 + y^2 = 1 \) in the line \( x + y = 1 \) is
(a) \( x^2 + y^2 + 2x + 2y + 1 = 0 \)
(b) \( x^2 + y^2 - 2x + 2y + 1 = 0 \)
(c) \( x^2 + y^2 + 2x - 2y + 1 = 0 \)
(d) \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
Answer: (d) \( x^2 + y^2 - 2x - 2y + 1 = 0 \)

Question. Image of (1,2) w.r.t. (-2,-1) is
(a) (0,5)
(b) (-4,-3)
(c) (-5,-4)
(d) (-4,-5)
Answer: (c) (-5,-4)

Question. The image of the point (-2,-7) under the transformation \( (x,y) \to (x-2y,-3x+y) \) is
(a) (–12,1)
(b) (12,–1)
(c) (–12,–1)
(d) (12,1)
Answer: (b) (12,–1)

Question. The algebraic sum of the perpendicular distances from the vertices of a triangle to a variable line is ‘0’, then the line passes through the ------ of the triangle
(a) Incentre
(b) Centroid
(c) Orthocentre
(d) Circumcentre
Answer: (b) Centroid

Question. A(1,-1) B(4,-1) C(4,3) are the vertices of a triangle. Then the equation of the altitude through the vertex ‘A’ is
(a) x = 4
(b) y = 4
(c) y + 1 = 0
(d) x = 1
Answer: (c) y + 1 = 0

Question. The equations of the sides of a triangle are \( x-3y=0, 4x+3y=5, 3x+y=0 \). The line \( 3x-4y=0 \) passes through
(a) Incentre
(b) Centroid
(c) Orthocentre
(d) Circumcentre
Answer: (c) Orthocentre

Question. Equation of a diameter of the circum circle of the triangle formed by the lines \( 3x+4y-7=0, 3x-y+5=0 \) and \( 8x-6y+1=0 \) is
(a) \( 3x-y-5=0 \)
(b) \( 3x+y+5=0 \)
(c) \( 3x-y+5=0 \)
(d) \( 3x+y-5=0 \)
Answer: (c) \( 3x-y+5=0 \)

Question. The incentre of the triangle formed by the lines \( x \cos \alpha + y \sin \alpha = \pi \), \( x \cos \beta + y \sin \beta = \pi \), \( x \cos \gamma + y \sin \gamma = \pi \) is \( (\alpha, \beta) \) then \( \alpha + \beta = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (a) 0

Question. The incentre of the triangle formed by the lines \( 3x + 4y = 10, 5x + 12y = 26, 7x + 24y = 50 \) is \( (\alpha, \beta) \) then \( \alpha + \beta = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (a) 0

Question. The lines \( p(p^2 + 1)x - y + q = 0 \) and \( (p^2 + 1)^2x + (p^2 + 1)y + 2q = 0 \) are perpendicular to a common line for
(a) exactly one value of p
(b) exactly two values of p
(c) more than two values of p
(d) no values of p
Answer: (a) exactly one value of p

Question. The slope of the line passing through the points \( (2, \sin \theta) \) and \( (1, \cos \theta) \) is 0 then general solution of \( \theta \)
(a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(b) \( n\pi - \frac{\pi}{4}, \forall n \in Z \)
(c) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(d) \( n\pi, \forall n \in Z \)
Answer: (a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)

Slope-intercept form, slope-point form and two-point form:

Question. The perpendicular bisector of the line segment joining \( P(1, 4) \) and \( Q(K, 3) \) has Y intercept -4. then a possible value of K is
(a) -4
(b) 1
(c) 2
(d) -2
Answer: (a) -4

Question. \( P(\alpha, \beta) \) lies on the line \( y = 6x - 1 \) and \( Q(\beta, \alpha) \) lies on the line \( 2x - 5y = 5 \). Then the equation of the line \( \overline{PQ} \) is
(a) \( 2x + y = 3 \)
(b) \( 3x + 2y = 5 \)
(c) \( x + y = 6 \)
(d) \( 3x + y = 7 \)
Answer: (c) \( x + y = 6 \)

Question. A line joining \( A(2, 0) \) and \( B(3, 1) \) is rotated about A in anticlockwise direction through angle \( 15^\circ \), then the equation of AB in the new position is
(a) \( y = \sqrt{3}x - 2 \)
(b) \( y = \sqrt{3}(x - 2) \)
(c) \( y = \sqrt{3}(x + 2) \)
(d) \( x - 2 = \sqrt{3}y \)
Answer: (b) \( y = \sqrt{3}(x - 2) \)

Intercepts and intercept form:

Question. The line \( 2x + 3y = 6, 2x + 3y = 8 \) cut the X-axis at A, B respectively. A line \( L = 0 \) drawn through the point \( (2, 2) \) meets the X-axis at C in such a way that abscissa of A, B, C are in arithmetic Progression. then the equation of the line L is
(a) \( 2x + 3y = 10 \)
(b) \( 3x + 2y = 10 \)
(c) \( 2x - 3y = 10 \)
(d) \( 3x - 2y = 10 \)
Answer: (a) \( 2x + 3y = 10 \)

Question. The sum of the intercepts cut off by the axes on lines \( x + y = a, x + y = ar, x + y = ar^2, \dots \) where \( a \neq 0 \) and \( r = \frac{1}{2} \)
(a) \( 2a \)
(b) \( a\sqrt{2} \)
(c) \( 2\sqrt{2}a \)
(d) \( a \)
Answer: (c) \( 2\sqrt{2}a \)

Question. The equation of the straight line which bisects the intercepts between the axes of the lines \( x + y = 2 \) and \( 2x + 3y = 6 \) is
(a) \( 2x = 3 \)
(b) \( y = 1 \)
(c) \( 2y = 3 \)
(d) \( x = 1 \)
Answer: (b) \( y = 1 \)

Question. Equation of the line passing through \( (0, 1) \) and having intercepts in the ratio \( 2 : 3 \) is
(a) \( 2x + 3y = 3 \)
(b) \( 2x - 3y + 3 = 0 \)
(c) \( 3x + 2y = 2 \)
(d) \( 2x - 3y - 3 = 0 \)
Answer: (c) \( 3x + 2y = 2 \)

Normal form and symmetric form:

Question. A straight line is such that its distance of 5 units from the origin and its inclination is \( 135^\circ \). The intercepts of the line on the coordinate axes are
(a) 5, 5
(b) \( \sqrt{2}, \sqrt{2} \)
(c) \( 5\sqrt{2}, 5\sqrt{2} \)
(d) \( 5/\sqrt{2}, 5/\sqrt{2} \)
Answer: (c) \( 5\sqrt{2}, 5\sqrt{2} \)

Question. Angles made with the x - axis by two lines drawn through the point \( (1, 2) \) and cutting the line \( x + y = 4 \) at a distance \( \sqrt{\frac{2}{3}} \) from the point (1,2) are
(a) \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{8} \) and \( \frac{3\pi}{8} \)
(c) \( \frac{\pi}{12} \) and \( \frac{5\pi}{12} \)
(d) \( \frac{\pi}{4} \) and \( \frac{\pi}{2} \)
Answer: (c) \( \frac{\pi}{12} \) and \( \frac{5\pi}{12} \)

Problems on distances:

Question. Perpendicular distance from the origin to the line joining the points \( (a \cos \theta, a \sin \theta) \) and \( (a \cos \phi, a \sin \phi) \) is
(a) \( 2a \cos (\theta - \phi) \)
(b) \( a \cos \left( \frac{\theta - \phi}{2} \right) \)
(c) \( 4a \cos \left( \frac{\theta - \phi}{2} \right) \)
(d) \( a \cos \left( \frac{\theta + \phi}{2} \right) \)
Answer: (b) \( a \cos \left( \frac{\theta - \phi}{2} \right) \)

Question. One side of an equilateral triangle is \( 3x + 4y = 7 \) and its vertex is \( (1, 2) \). Then the length of the side of the triangle is
(a) \( \frac{4\sqrt{3}}{17} \)
(b) \( \frac{3\sqrt{3}}{16} \)
(c) \( \frac{8\sqrt{3}}{15} \)
(d) \( \frac{4\sqrt{3}}{15} \)
Answer: (c) \( \frac{8\sqrt{3}}{15} \)

Question. Equation of the line through the point of intersection of the lines \( 3x + 2y + 4 = 0 \) and \( 2x + 5y - 1 = 0 \) whose distance from \( (2, -1) \) is 2.
(a) \( 2x - y + 5 = 0 \)
(b) \( 4x + 3y + 5 = 0 \)
(c) \( x + 2 = 0 \)
(d) \( 3x + y + 5 = 0 \)
Answer: (b) \( 4x + 3y + 5 = 0 \)

Question. If p, q denote the lengths of the perpendiculars from the origin on the lines \( x \sec \alpha - y \csc \alpha = a \) and \( x \cos \alpha + y \sin \alpha = a \cos 2\alpha \) then
(a) \( 4p^2 + q^2 = a^2 \)
(b) \( p^2 + q^2 = a^2 \)
(c) \( p^2 + 2q^2 = a^2 \)
(d) \( 4p^2 + q^2 = 4a^2 \)
Answer: (a) \( 4p^2 + q^2 = a^2 \)

Question. The distance between two parallel lines is \( p_1 - p \). If equation of one line is \( x \cos \alpha + y \sin \alpha = p \) then the equation of the 2nd line is
(a) \( x \cos \alpha + y \sin \alpha + p_1 + 2p = 0 \)
(b) \( x \cos \alpha + y \sin \alpha = 2p_1 - p \)
(c) \( x \cos \alpha + y \sin \alpha = 0 \)
(d) \( x \cos \alpha + y \sin \alpha + p_1 - 2p = 0 \)
Answer: (d) \( x \cos \alpha + y \sin \alpha + p_1 - 2p = 0 \)

Question. The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is
(a) 7 : 3
(b) 3 : 7
(c) 2 : 3
(d) 3 : 4
Answer: (b) 3 : 7

Question. The equations of the lines parallel to \( 4x + 3y + 2 = 0 \) and at a distance of '4' units from it are
(a) \( 4x + 3y + 22 = 0, 4x + 3y - 20 = 0 \)
(b) \( 4x + 3y + 22 = 0, 4x + 3y - 18 = 0 \)
(c) \( 4x + 3y - 18 = 0, 4x + 3y - 20 = 0 \)
(d) \( 4x - 3y - 18 = 0, 4x + 3y - 20 = 0 \)
Answer: (b) \( 4x + 3y + 22 = 0, 4x + 3y - 18 = 0 \)

Position of a point (s) w.r.t. line (s):

Question. The range of \( \alpha \) for which the points \( (\alpha, \alpha + 2) \) and \( \left( \frac{3\alpha}{2}, \alpha^2 \right) \) lie on opposite sides of the line \( 2x + 3y - 6 = 0 \)
(a) \( (-\infty, -2) \)
(b) \( (0, 1) \)
(c) \( (-\infty, -2) \cup (0, 1) \)
(d) \( (-\infty, 1) \cup (2, \infty) \)
Answer: (c) \( (-\infty, -2) \cup (0, 1) \)

Question. If \( P \left( 1 + \frac{t}{\sqrt{2}}, 2 + \frac{t}{\sqrt{2}} \right) \) be any point on a line then the range of values of t for which the point P lies between the parallel lines \( x + y = 1 \) and \( 2x + 2y = 15 \) is
(a) \( \frac{-4\sqrt{2}}{5} < t < \frac{5\sqrt{2}}{6} \)
(b) \( \frac{-4\sqrt{2}}{3} < t < \frac{5\sqrt{2}}{6} \)
(c) \( t < \frac{-4\sqrt{2}}{3} \)
(d) \( t < \frac{5\sqrt{2}}{6} \)
Answer: (b) \( \frac{-4\sqrt{2}}{3} < t < \frac{5\sqrt{2}}{6} \)

Question. A point which lies between \( 2x + 3y - 7 = 0 \) and \( 2x + 3y + 12 = 0 \) is
(a) \( (5, 1) \)
(b) \( (-1, 3) \)
(c) \( (3, -5) \)
(d) \( (7, -1) \)
Answer: (c) \( (3, -5) \)

Question. A line L cuts the sides AB, BC of \( \Delta ABC \) in the ratio 2 : 5, 7 : 4 respectively. Then the line L cuts CA in the ratio
(a) 7 : 10
(b) 7 : -10
(c) 10 : 7
(d) 10 : -7
Answer: (d) 10 : -7

Question. The number of integral values of m for which x-coordinate of point of intersection of the lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is also an integer is
(a) 2
(b) 0
(c) 4
(d) 11
Answer: (a) 2