Class 11 Mathematics Straight Lines MCQs Set 17

Slope of a line:

Question. If the inclination of the line \( (2-k)x - (1-k)y + (5-2k) = 0 \) is \( \frac{3\pi}{4} \) then the value of k is
(a) \( \frac{5}{2} \)
(b) \( -\frac{3}{2} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (d) \( \frac{3}{2} \)

Question. If the line joining the points \( (at_1^2, 2at_1) \) and \( (at_2^2, 2at_2) \) is parallel to \( y = x \) then \( t_1 + t_2 = \)
(a) \( \frac{1}{2} \)
(b) 4
(c) \( \frac{1}{4} \)
(d) 2
Answer: (d) 2

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

Question. The line \( 2x + 3y + 12 = 0 \) cuts the axes at A & B. Then the equation of the perpendicular bisector of \( \overline{AB} \) is
(a) \( 3x - 2y + 5 = 0 \)
(b) \( 3x - 2y + 7 = 0 \)
(c) \( 3x - 2y + 9 = 0 \)
(d) \( 3x - 2y + 8 = 0 \)
Answer: (a) \( 3x - 2y + 5 = 0 \)

Question. If \( t_1, t_2 \) are roots of the equation \( t^2 + \lambda t + 1 = 0 \) where \( \lambda \) is an arbitrary constant, then the line joining the points \( (at_1^2, 2at_1) \), \( (at_2^2, 2at_2) \) always passes through the fixed point
(a) \( (-a, 0) \)
(b) \( (0, a) \)
(c) \( (a, 0) \)
(d) \( (0, -a) \)
Answer: (a) \( (-a, 0) \)

Question. ABCD is a parallelogram. Equations of AB and AD are \( 4x + 5y = 0 \) and \( 7x + 2y = 0 \) and the equation of diagonal BD is \( 11x + 7y + 9 = 0 \). The equation of AC is
(a) \( x + y = 0 \)
(b) \( x - y = 0 \)
(c) \( x + y + 1 = 0 \)
(d) \( x + y - 1 = 0 \)
Answer: (b) \( x - y = 0 \)

Intercepts and intercept form:

Question. The equation of the straight line whose intercepts on x-axis and y-axis are respectively twice and thrice of those by the line \( 3x + 4y = 12 \), is
(a) \( 9x + 8y = 72 \)
(b) \( 9x - 8y = 72 \)
(c) \( 8x + 9y = 72 \)
(d) \( 8x + 9y + 72 = 0 \)
Answer: (a) \( 9x + 8y = 72 \)

Question. The equation of a straight line parallel to \( 2x + 3y + 11 = 0 \) and which is such that the sum of its intercepts on the axes is 15.
(a) \( 2x + 3y = 15 \)
(b) \( 3x + 2y = 10 \)
(c) \( 2x - 3y = 10 \)
(d) \( 2x + 3y = 18 \)
Answer: (d) \( 2x + 3y = 18 \)

Question. The straight line through \( P(1, 2) \) is such that its intercept between the axes is bisected at P. Its equation is
(a) \( x + 2y = 5 \)
(b) \( x + y - 3 = 0 \)
(c) \( x - y + 1 = 0 \)
(d) \( 2x + y - 4 = 0 \)
Answer: (d) \( 2x + y - 4 = 0 \)

Question. If \( (4, -3) \) divides the line segment between the axes in the ratio \( 4 : 5 \) then its equation is
(a) \( 15x + 16y - 12 = 0 \)
(b) \( 3x - 4y - 24 = 0 \)
(c) \( 15x - 16y + 108 = 0 \)
(d) \( 15x - 16y - 108 = 0 \)
Answer: (d) \( 15x - 16y - 108 = 0 \)

Normal form and symmetric form:

Question. If a line AB makes an angle \( \theta \) with OX and is at a distance of p units from the origin then the equation of AB is
(a) \( x \sin \theta - y \cos \theta = p \)
(b) \( x \cos \theta + y \sin \theta = p \)
(c) \( x \sin \theta + y \cos \theta = p \)
(d) \( x \cos \theta - y \sin \theta = p \)
Answer: (a) \( x \sin \theta - y \cos \theta = p \)

Question. The parametric equation of a line is given by \( x = -2 + \frac{r}{\sqrt{10}} \) and \( y = 1 + \frac{3r}{\sqrt{10}} \). Then, for the line
(a) intercept on the \( x\text{-axis} = -\frac{7}{3} \)
(b) intercept on the \( y\text{-axis} = -7 \)
(c) slope of the line = 1/3
(d) slope of the line = 3
Answer: (d) slope of the line = 3

Problems on distances:

Question. A straight line through the origin 'O' meets the parallel lines \( 4x + 2y = 9 \) and \( 2x + y + 6 = 0 \) at points P and Q respectively, then point O divides the segment PQ in the ratio
(a) 1 : 2
(b) 3 : 4
(c) 2 : 1
(d) 4 : 3
Answer: (b) 3 : 4

Question. The lengths of the perpendiculars from \( (m^2, 2m) \), \( (mn, m + n) \) and \( (n^2, 2n) \) to the straight line \( x \cos \alpha + y \sin \alpha + \sin \alpha \tan \alpha = 0 \) are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P.
Answer: (b) G.P.

Question. The distance between the Straight lines \( y = mx + c_1 \), \( y = mx + c_2 \) is \( |c_1 - c_2| \) then m =
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0

Question. Distance between parallel lines \( 4x + 6y + 8 = 0 \), \( 6x + 9y + 15 = 0 \) is
(a) \( 2 / \sqrt{13} \)
(b) \( 1 / \sqrt{13} \)
(c) \( 3 / \sqrt{13} \)
(d) \( 4 / \sqrt{13} \)
Answer: (b) \( 1 / \sqrt{13} \)

Question. \( 2x + 3y - 5 = 0 \), \( 2x + 3y + 15 = 0 \), \( x + y - 7 = 0 \), \( x + y + 7 = 0 \) are sides of a parallelogram. Then the centre of the parallelogram is
(a) \( (-5, -5) \)
(b) \( (5, -5) \)
(c) \( (-5, 5) \)
(d) \( (5, 5) \)
Answer: (b) \( (5, -5) \)

Question. The distance of the point \( (3, 5) \) from the line \( 2x + 3y - 14 = 0 \) measured parallel to the line \( x - 2y = 1 \) is
(a) \( \frac{7}{\sqrt{5}} \)
(b) \( \frac{7}{\sqrt{13}} \)
(c) \( \sqrt{5} \)
(d) \( \sqrt{13} \)
Answer: (c) \( \sqrt{5} \)

Question. Equation of the straight line passing through \( (1,1) \) and at a distance of 3 units from \( (-2, 3) \) is
(a) \( x - 2 = 0 \)
(b) \( 5x - 12y + 6 = 0 \)
(c) \( 5x - 12y + 7 = 0 \)
(d) \( y - 1 = 0 \)
Answer: (c) \( 5x - 12y + 7 = 0 \)

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

Question. If \( L_1, L_2 \) denote the lines \( x + 2y - 2 = 0 \), \( 2x + 3y + 4 = 0 \)
(a) \( L_1 \) is nearer to origin than \( L_2 \)
(b) \( L_2 \) is nearer to origin than \( L_1 \)
(c) \( L_1, L_2 \) are equidistant from origin
(d) can’t say
Answer: (a) \( L_1 \) is nearer to origin than \( L_2 \)

Question. If the point \( (a, a) \) falls between the lines \( |x+y| = 2 \), then:
(a) \( |a| = 2 \)
(b) \( |a| = 1 \)
(c) \( |a| < 1 \)
(d) \( |a| < 1/2 \)
Answer: (c) \( |a| < 1 \)

Question. If \( (2a-3, a^2-1) \) is on the same side of the line \( x + y - 4 = 0 \) as that of origin then the set of values of ‘a’ is
(a) \( (-4, 2) \)
(b) \( (-2, 4) \)
(c) \( (-7, 8) \)
(d) \( (-7, 5) \)
Answer: (a) \( (-4, 2) \)

Question. The range of values of the ordinate of a point moving on the line \( x=1 \) and always remaining in the interior of the triangle formed by the lines \( y = x \), the x-axis and \( x+y=4 \)
(a) \( (0, 1) \)
(b) \( (0, 2) \)
(c) \( (1, 2) \)
(d) \( (2, 1) \)
Answer: (a) \( (0, 1) \)

Point of intersection of lines and concurrency of Lines :

Question. If the line \( \frac{x-1}{2} = \frac{y-2}{3} = t \) intersects the line \( x+y=8 \) then \( t = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Equation of line which is equally inclined to the axis and passes through a common points of family of lines \( 4acx + y(ab + bc + ca – abc) + abc = 0 \) (where \( a, b, c > 0 \) are in H.P.) is
(a) \( y - x = \frac{7}{4} \)
(b) \( y + x = \frac{7}{4} \)
(c) \( y - x = \frac{1}{4} \)
(d) \( y + x = \frac{3}{4} \)
Answer: (a) \( y - x = \frac{7}{4} \)

Question. If a,b,c in GP then the line \( a^2x + b^2y + ac = 0 \) always passes through the fixed point
(a) \( (0, 1) \)
(b) \( (1, 0) \)
(c) \( (0, -1) \)
(d) \( (1, -1) \)
Answer: (c) \( (0, -1) \)

Question. If \( U \equiv x+y-2 = 0 \), \( V \equiv 2x-3y+1 = 0 \), the point of intersection of the lines \( 50U+7V=0 \), \( 3U+11V=0 \) is
(a) \( (0, 0) \)
(b) \( (1, 0) \)
(c) \( (0, 1) \)
(d) \( (1, 1) \)
Answer: (d) \( (1, 1) \)

Question. The straight lines \( x+2y-9=0 \), \( 3x+5y-5=0 \) and \( ax+by-1=0 \) are concurrent if the straight line \( 22x-35y-1=0 \) passes through the point
(a) \( (a, b) \)
(b) \( (b, a) \)
(c) \( (-a, b) \)
(d) \( (-a, -b) \)
Answer: (b) \( (b, a) \)

Question. The equation of the line passing through the point of intersection of the lines \( 2x+y=5 \) and \( y=3x-5 \) and which is at the minimum distance from the point \( (1,2) \) is
(a) \( x+y=3 \)
(b) \( x-y=1 \)
(c) \( x-2y=0 \)
(d) \( 2x+5y=7 \)
Answer: (a) \( x+y=3 \)

Question. Given a family of lines \( a(2x + y + 4) + b(x – 2y – 3) = 0 \). The number of lines belonging to the family at a distance \( \sqrt{10} \) from \( P(2, –3) \) is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (b) 1

Angle between lines:

Question. The acute angle between the lines \( lx + my = l+m \), \( l(x-y) + m(x+y) = 2m \) is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{\pi}{3} \)
Answer: (a) \( \frac{\pi}{4} \)

Question. The angle between the lines \( x \cos \alpha + y \sin \alpha = p_1 \) and \( x \cos \beta + y \sin \beta = p_2 \) where \( \alpha > \beta \) is
(a) \( \alpha + \beta \)
(b) \( \alpha - \beta \)
(c) \( \alpha \beta \)
(d) \( 2\alpha - \beta \)
Answer: (b) \( \alpha - \beta \)

Question. One vertex of an equilateral triangle is \( (2,3) \) and the equation of one side is \( x-y+5=0 \). Then the equations to other sides are
(a) \( y – 3 = –(2 \pm \sqrt{3})(x – 2) \)
(b) \( y – 3 = (\sqrt{2} \pm 1)(x – 2) \)
(c) \( y – 3 = (\sqrt{3} \pm 1)(x – 2) \)
(d) \( y – 3 = (\sqrt{5} \pm 1)(x – 2) \)
Answer: (a) \( y – 3 = –(2 \pm \sqrt{3})(x – 2) \)

Question. Two equal sides of an isosceles triangle are given by the equations \( 7x – y + 3 = 0 \) and \( x + y – 3 = 0 \). The slope of the third side is
(a) \( 3, -1/3 \)
(b) \( 3, 1/3 \)
(c) \( -3, 1/3 \)
(d) \( -3, -1/3 \)
Answer: (a) \( 3, -1/3 \)

Question. Let there are two lines \( 2x + 3y + \lambda = 0 \) and \( \lambda x - 3y - 1 = 0 \). If the origin lies in the obtuse angle then
(a) \( \lambda = \frac{9}{2} \)
(b) \( -2 < \lambda < 0 \)
(c) \( 0 < \lambda < \frac{9}{2} \)
(d) None of the options
Answer: (c) \( 0 < \lambda < \frac{9}{2} \)

Triangles, area of the triangle:

Question. The area of the triangle formed by the axes and the line \( (\cosh \alpha - \sinh \alpha)x + (\cosh \alpha + \sinh \alpha)y = 2 \) in square units is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (c) 2

Question. The equation to the base of an equilateral triangle is \( (\sqrt{3} + 1)x + (\sqrt{3} - 1)y + 2\sqrt{3} = 0 \) and opposite vertex is \( A(1, 1) \) then the Area of the triangle is
(a) \( 3\sqrt{2} \)
(b) \( 3\sqrt{3} \)
(c) \( 2\sqrt{3} \)
(d) \( 4\sqrt{3} \)
Answer: (c) \( 2\sqrt{3} \)

Question. Equation of the line on which the perpendicular from the origin makes an angle of \( 30^\circ \) with X-axis and which forms a triangle of area \( \frac{50}{\sqrt{3}} \) with the axes is
(a) \( \sqrt{2}x + 2y = 9 \)
(b) \( \sqrt{2}x + 3y = \pm 9 \)
(c) \( \sqrt{3}x - y = \pm 10 \)
(d) \( \sqrt{3}x + y = \pm 10 \)
Answer: (d) \( \sqrt{3}x + y = \pm 10 \)

Quadrilaterals and area of the quadrilaterals:

Question. The point \( (2,3) \) is reflected four times about co-ordinate axes continuously starting with x-axis. The area of quadrilateral formed in sq.units is
(a) 24
(b) 6
(c) 12
(d) 5
Answer: (a) 24

Question. Area of the quadrilateral formed by the lines \( 4y - 3x - a = 0 \), \( 3y - 4x + a = 0 \), \( 4y - 3x - 3a = 0 \), \( 3y - 4x + 2a = 0 \) is
(a) \( \frac{a^2}{5} \)
(b) \( \frac{a^2}{7} \)
(c) \( \frac{2a^2}{7} \)
(d) \( \frac{2a^2}{9} \)
Answer: (c) \( \frac{2a^2}{7} \)

Question. Two sides of a rectangle are \( 3x + 4y + 5 = 0 \), \( 4x - 3y + 15 = 0 \) and its one vertex is \( (0,0) \). Then the area of the rectangle is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (b) 3