Class 11 Mathematics Straight Lines MCQs Set 13

Question. The straight line through A(a,b) intersects the line through B(c,d) at ‘P’ at right angles. The locus of P is
(a) \( (x-a)(x-c)+(y-b)(y-d)=0 \)
(b) \( (x-a)(x-c)-(y-b)(y-d)=0 \)
(c) \( (x-b)(x-d)+(y-a)(y-d)=0 \)
(d) \( (x-b)(x-d)+(y-a)(y-c)=0 \)
Answer: (a) \( (x-a)(x-c)+(y-b)(y-d)=0 \)

Question. If \( ax+by+c=0 \) is parallel to x-axis then which of the following is defined
(a) \( \frac{a^2-c^2}{c^2+b^2} \)
(b) \( \frac{b^2-c^2}{a^2} \)
(c) \( \frac{4b^2+c^2}{\sqrt{abc}} \)
(d) \( \frac{\sqrt{a+c^2}}{a} \)
Answer: (a) \( \frac{a^2-c^2}{c^2+b^2} \)

Question. The straightline \( ax+by+c=0 \, (a,b,c \neq 0) \) will pass through the first quadrant and cut the positive x-axis, if
(a) \( ac > 0, bc > 0 \)
(b) \( ac > 0, bc < 0 \)
(c) \( ac > 0 \) and / or \( bc > 0 \)
(d) \( ac < 0 \) and / or \( bc < 0 \)
Answer: (d) \( ac < 0 \) and / or \( bc < 0 \)

Question. If \( \frac{ax}{\sqrt{a^2+b^2}} + \frac{by}{\sqrt{a^2+b^2}} = \frac{-c}{\sqrt{a^2+b^2}} \) is perpendicular form of a straight line then
(a) \( a, b, c > 0 \)
(b) \( a > 0, b > 0, c < 0 \)
(c) \( c < 0 \)
(d) \( a, b > 0, c \neq 0 \)
Answer: (c) \( c < 0 \)

Question. The straight line passing through \( P(x_1, y_1) \) and making an angle \( \alpha \) with x-axis intersects \( Ax+By+C=0 \) in Q then PQ=
(a) \( \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \)
(b) \( \frac{|Ax_1 + By_1 + C|}{A \cos \alpha + B \sin \alpha} \)
(c) \( \frac{|Ax_1 + By_1 + C|}{A \cos \alpha + B \sin \alpha} \)
(d) \( \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 \cos^2 \alpha + B^2 \sin^2 \alpha}} \)
Answer: (c) \( \frac{|Ax_1 + By_1 + C|}{A \cos \alpha + B \sin \alpha} \)

Question. If \( a+b+c \neq 0, ax+by+c=0, bx+cy+a=0, cx+ay+b=0 \) are concurrent then \( \frac{a^2+b^2+c^2}{ab+bc+ca} = \)
(a) 1/2
(b) 2
(c) 1
(d) 0
Answer: (c) 1

Question. The lines \( (a+b-2c)x+(b+c-2a)y+(c+a-2b)=0 \), \( (b+c-2a)x+(c+a-2b)y+(a+b-2c)=0 \) and \( (c+a-2b)x+(a+b-2c)y+(b+c-2a)=0 \) where a,b,c are real numbers
(a) Form an equilateral triangle
(b) Concurrent
(c) Form a right angled triangle
(d) Form an isosceles triangle
Answer: (b) Concurrent

Question. If the lines \( p_1x+q_1y=1, p_2x+q_2y=1 \) and \( p_3x+q_3y=1 \) be concurrent, then the points \( (p_1, q_1), (p_2, q_2) \) and \( (p_3, q_3) \)
(a) are collinear
(b) form an equilateral triangle
(c) form a scalene triangle
(d) form a right angled triangle
Answer: (a) are collinear

Question. \( m \equiv a_1x+b_1y+c_1=0 \) and \( l \equiv a_2x+b_2y+c_2=0 \) are two straight lines such that \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) then \( m + kl = 0, k \in R \) is
(a) a straight line different from m and l
(b) not a straight line
(c) is a straight line concurrent with m and l
(d) the same straight line m = 0
Answer: (d) the same straight line m = 0

Question. If a and b are the intercepts made by the straight line on the coordinate axes such that \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} \) then the line passes through point
(a) (1,1)
(b) (c,c)
(c) \( (\frac{1}{c}, \frac{1}{c}) \)
(d) \( (\frac{c}{a}, \frac{c}{a}) \)
Answer: (b) (c,c)

Question. The orthocentre of the triangle formed by the points \( A(a \cos \alpha, a \sin \alpha) \), \( B(a \cos \beta, a \sin \beta) \), \( C(a \cos \gamma, a \sin \gamma) \) is
(a) \( (\cos \alpha + \cos \beta + \cos \gamma, \sin \alpha + \sin \beta + \sin \gamma) \)
(b) \( [a(\cos \alpha + \cos \beta + \cos \gamma), a(\sin \alpha + \sin \beta + \sin \gamma)] \)
(c) \( [a(\cos \alpha + \sin \beta + \sin \gamma), a(\sin \alpha + \cos \beta + \cos \gamma)] \)
(d) \( (\cos \alpha \cos \beta \cos \gamma, \sin \alpha \sin \beta \sin \gamma) \)
Answer: (b) \( [a(\cos \alpha + \cos \beta + \cos \gamma), a(\sin \alpha + \sin \beta + \sin \gamma)] \)

Question. (a, b), (c, d), (e, f) are the vertices of an equilateral triangle. Then the orthocentre of the triangle is
(a) \( \left( \frac{a+d+f}{3}, \frac{b+c+e}{3} \right) \)
(b) \( \left( \frac{a+c+e}{3}, \frac{b+d+f}{3} \right) \)
(c) \( \left( \frac{a+c+f}{3}, \frac{b+d+e}{3} \right) \)
(d) \( \left( \frac{a+b+c}{3}, \frac{d+e+f}{3} \right) \)
Answer: (b) \( \left( \frac{a+c+e}{3}, \frac{b+d+f}{3} \right) \)

Question. A triangle is formed by the lines \( ax+by+c=0 \), \( lx+my+n=0, px+qy+r=0 \), then the straight line \( (ax+by+c)(lp+mq) = (lx+my+n)(ap+bq) \) passes through .......... of the triangle.
(a) Incentre
(b) Circumcentre
(c) Orthocentre
(d) Centroid
Answer: (c) Orthocentre

Question. The area of the triangle formed by the lines \( y = m_1x + c_1, y = m_2x+c_2 \) and \( x=0 \) is
(a) \( \frac{|c_1 - c_2|}{2|m_1 - m_2|} \)
(b) \( \frac{(c_1 - c_2)^2}{2(m_2 - m_1)^2} \)
(c) \( \frac{(c_1 - c_2)^2}{2|m_1 - m_2|} \)
(d) \( \frac{(c_1 - c_2)^2}{|m_2 - m_1|} \)
Answer: (c) \( \frac{(c_1 - c_2)^2}{2|m_1 - m_2|} \)

Question. If the line passing through the points (-8,3) (2,1) is parallel to the line passing through the points (11,-1) (k,0) then the value of k is
(a) 5
(b) 7
(c) 5/2
(d) 6
Answer: (d) 6

Question. If each of the points (a,4),(-2,b) lies on the line joining the points (2,-1),(5,-3), then the point (a,b) lies on line
(a) \( 6x+6y-25=0 \)
(b) \( x+3y+1=0 \)
(c) \( 2x+6y+1=0 \)
(d) \( 2x+3y-5=0 \)
Answer: (c) \( 2x+6y+1=0 \)

Question. If the lines \( y = -3x + 4 \), \( ay = x + 10 \) and \( 2y + bx + 9 = 0 \) represent three consecutive sides of a rectangle then ab =
(a) 18
(b) -3
(c) 1/2
(d) -1/3
Answer: (a) 18

Question. The equation of the straight line cutting off an intercept 8 on x-axis and making an angle of \( 60^\circ \) with the positive direction of y-axis is
(a) \( x - \sqrt{3}y - 8 = 0 \)
(b) \( x + \sqrt{3}y = 8 \)
(c) \( y - \sqrt{3}x = 8 \)
(d) \( y + \sqrt{3}x = 8 \)
Answer: (b) \( x + \sqrt{3}y = 8 \)

Question. If (-4,5) is one vertex and \( 7x-y+8=0 \) is one diagonal of a square, then the equation of the other diagonal is
(a) \( x+7y-31=0 \)
(b) \( x+7y-15=0 \)
(c) \( x+7y+8=0 \)
(d) \( x+7y-35=0 \)
Answer: (a) \( x+7y-31=0 \)

Question. Equation of a line which passes through the point (-3,8) and cut off positive intercepts on the axes whose sum is 7 is
(a) \( 3x-4y=12 \)
(b) \( 4x+3y=12 \)
(c) \( 3x+4y=12 \)
(d) \( 4x-3y=12 \)
Answer: (b) \( 4x+3y=12 \)

Question. The number of lines that are parallel to \( 2x + 6y - 7 = 0 \) and have an intercept 10 between the co-ordinate axes is
(a) 1
(b) 2
(c) 4
(d) infinitely many
Answer: (b) 2

Question. If the line \( (x-y+1) + k(y-2x+4) = 0 \) makes equal intercept on the axes then the value of k is
(a) 1/3
(b) 3/4
(c) 1/2
(d) 2/3
Answer: (d) 2/3

Question. The equation of set of lines which are at a constant distance 2 units from the origin is
(a) \( x+y+2=0 \)
(b) \( x+y+4=0 \)
(c) \( x \cos \alpha + y \sin \alpha = 2 \)
(d) \( x \cos \alpha + y \sin \alpha = 1/2 \)
Answer: (c) \( x \cos \alpha + y \sin \alpha = 2 \)

Question. The slope of a straight line through A(3,2) is 3/4 then the coordinates of the two points on the line that are 5 units away from A are
(a) (-7,5) (1,-1)
(b) (7,5) (-1,-1)
(c) (6,9) (-2,4)
(d) (7,3) (-2,1)
Answer: (b) (7,5) (-1,-1)

Question. The length of the perpendicular from the point (0,0) to the straight line passing through P(1,2) such that P bisects the intercepted part between axes
(a) \( \sqrt{5} \)
(b) 4
(c) \( 4/\sqrt{5} \)
(d) \( \sqrt{5}/4 \)
Answer: (c) \( 4/\sqrt{5} \)

Question. Radius of the circle touching the lines \( 3x+4y-14=0 \), \( 6x+8y+7=0 \) is
(a) 7
(b) 7/2
(c) 7/4
(d) 7/6
Answer: (c) 7/4

Question. The distance between the parallel lines given by \( (x+y)^2 + 4\sqrt{2}(x+y) - 42 = 0 \) is
(a) 1
(b) 5
(c) 6
(d) 2
Answer: (d) 2

Question. Equation of the straight line parallel to \( x+2y-5=0 \) and at the same distance from (3,2) is
(a) \( x+2y-8=0 \)
(b) \( x+2y+9=0 \)
(c) \( x+2y-9=0 \)
(d) \( x+2y-7=0 \)
Answer: (c) \( x+2y-9=0 \)

Question. If the straight line drawn through the point \( P(\sqrt{3}, 2) \) making an angle \( \frac{\pi}{6} \) with x-axis meets the line \( \sqrt{3}x-4y+8=0 \) at Q. Then PQ is
(a) 4
(b) 5
(c) 6
(d) 9
Answer: (c) 6

Question. If the line \( 3x+4y-8=0 \) is denoted by L, then the points (2,-5),(-5,2)
(a) lie on L
(b) lie on same side of L
(c) lie on opposite sides of L
(d) equidistant from L
Answer: (b) lie on same side of L

Question. The vertices of a triangle are O(0.0), B(-3,-1),C(-1,-3). The equation of the line parallel to BC and intersecting the sides OB and OC whose perpendicular distance from O is 1/2 is
(a) \( x + y = 1/\sqrt{2} \)
(b) \( x + y = -1/\sqrt{2} \)
(c) \( x + y = -1/2 \)
(d) \( x + y = 1/2 \)
Answer: (b) \( x + y = -1/\sqrt{2} \)

Question. If the lines \( ax+by+c = 0 \), \( bx+cy+a = 0 \) and \( cx+ay+b=0 \, (a \neq b \neq c) \) are concurrent then the point of concurrency is
(a) (0,0)
(b) (1,1)
(c) (2,2)
(d) (-1,-1)
Answer: (b) (1,1)

Question. If the lines \( 3x+2y-5=0 \), \( 2x-5y+3=0 \), \( 5x+by+c=0 \) are concurrent then b+c =
(a) 7
(b) -5
(c) 6
(d) 9
Answer: (b) -5

Question. The lines \( (p-q)x + (q-r)y + (r-p) = 0 \), \( (q-r)x + (r-p)y + (p-q) = 0 \), \( (r-p)x + (p-q)y + (q-r) = 0 \) are
(a) Form an equilateral triangle
(b) Form an Isosceles triangle
(c) Concurrent
(d) Form a right angled triangle
Answer: (c) Concurrent

Question. If 2 is a root of \( ax^2+bx+c=0 \) then point of concurrence of lines \( ax+2by+3c=0 \) is
(a) (12,3)
(b) (4,2)
(c) (1,2)
(d) (2,3)
Answer: (a) (12,3)

Question. For all values of ‘a’ the set of straight lines \( (3a+1)x - (2a+3)y + 9-a=0 \) passes through the point
(a) (3, 4)
(b) (4,2)
(c) (3,3)
(d) (1,2)
Answer: (a) (3, 4)

Question. Equation of the line passing through the point of intersection of the lines \( 2x+3y-1=0 \), \( 3x+4y-6=0 \) and perpendicular to \( 5x-2y-7=0 \) is
(a) \( 2x+5y-19=0 \)
(b) \( 2x+5y+17=0 \)
(c) \( 2x+5y-16=0 \)
(d) \( 2x+5y-22=0 \)
Answer: (b) \( 2x+5y+17=0 \)

Question. If \( 2x + 3y + 4 = 0 \) & \( \lambda x + ky + 2 = 0 \) are identical lines then \( 3\lambda - 2k = \)
(a) 1
(b) 0
(c) -1
(d) 2
Answer: (b) 0

Question. The angle between the diagonals of a quadrilateral formed by the lines \( \frac{x}{a} + \frac{y}{b} = 1 \), \( \frac{x}{b} + \frac{y}{a} = 1 \), \( \frac{x}{a} + \frac{y}{b} = 2 \), \( \frac{x}{b} + \frac{y}{a} = 2 \) is
(a) \( \pi/4 \)
(b) \( \pi/6 \)
(c) \( \pi/3 \)
(d) \( \pi/2 \)
Answer: (d) \( \pi/2 \)

Question. The triangle formed by the lines \( \sqrt{3}x+y-2=0 \), \( \sqrt{3}x-y+1=0 \), \( y=0 \) is
(a) Equilateral
(b) Right angled
(c) Right angled isosceles
(d) Isosceles
Answer: (a) Equilateral