Class 11 Mathematics Straight Lines MCQs Set 11

Question. If \( \frac{a}{\sqrt{bc}} - 2 = \sqrt{\frac{b}{c}} + \sqrt{\frac{c}{b}} \) where \( a, b, c > 0 \) then family of lines \( \sqrt{a}x + \sqrt{b}y + \sqrt{c} = 0 \) passes through the point
(a) \( (1,1) \)
(b) \( (1, -2) \)
(c) \( (-1, 2) \)
(d) \( (-1, 1) \)
Answer: (d) \( (-1, 1) \)

Question. The number of integer values of \( \lambda \) for which the x-coordinates of the point of intersection of the lines \( 3x + 4y = 9 \) and \( y = \lambda x + 1 \) is also an integer is \( m \). Then the area between the lines \( y = mx + r \), \( r = 1, 2 \) and bounded by coordinate axes is (in sq.units)
(a) \( \frac{3}{4} \)
(b) \( \frac{4}{9} \)
(c) \( \frac{1}{3} \)
(d) \( \frac{4}{3} \)
Answer: (a) \( \frac{3}{4} \)

Question. Let \( 2a + 3b + c = 0 \). If \( x + 5y + 7 = 0 \) and \( L = 0 \) are angle bisectors of two lines \( L_1 = 0, L_2 = 0 \) and \( L = 0 \) is a member in the family of lines \( ax + by + c = 0 \) then area of triangle formed by \( L = 0 \) with coordinate axes is (in sq.units)
(a) 49
(b) \( \frac{49}{2} \)
(c) \( \frac{49}{5} \)
(d) \( \frac{49}{10} \)
Answer: (d) \( \frac{49}{10} \)

Question. The line \( 2x + y = 4 \) meet x-axis at \( A \) and y-axis at \( B \). The perpendicular bisector of \( AB \) meets the horizontal line through \( (0, -1) \) at \( C \). Let \( G \) be the centroid of \( \Delta ABC \). The perpendicular distance from \( G \) to \( AB \) equals
(a) \( \sqrt{5} \)
(b) \( \frac{\sqrt{5}}{3} \)
(c) \( 2\sqrt{5} \)
(d) \( 3\sqrt{5} \)
Answer: (a) \( \sqrt{5} \)

Question. The distance of any point \( (x, y) \) from the origin is defined as \( d = \max \{ |x|, |y| \} \), then the distance of the common point for the family of lines \( x(1 + \lambda) + \lambda y + 2 + \lambda = 0 \) (\( \lambda \) being parameter) from origin is
(a) 1
(b) 2
(c) \( \sqrt{5} \)
(d) 0
Answer: (b) 2

Question. The distance between two parallel lines is 1 unit. A point ‘A’ is chosen to lie between the lines at distance ‘d’ from one of them. Triangle \( ABC \) is equilateral with \( B \) on one line and \( C \) on the other parallel line the length of the side of the equilateral triangle is (in units)
(a) \( (2/3)\sqrt{d^2 + d + 1} \)
(b) \( 2\sqrt{(d^2 - d + 1)/3} \)
(c) \( 2\sqrt{(d^2 - d + 1)} \)
(d) \( \sqrt{(d^2 - d + 1)} \)
Answer: (b) \( 2\sqrt{(d^2 - d + 1)/3} \)

Question. If the point \( P(\alpha^2, \alpha) \) lies in the region corresponding to the acute angle between the lines \( x - 3y = 0 \) and \( x - 5y = 0 \) then
(a) \( \alpha \in (5, 15) \)
(b) \( \alpha \in (5, 8) \)
(c) \( \alpha \in (4, 8) \)
(d) \( \alpha \in (3, 5) \)
Answer: (d) \( \alpha \in (3, 5) \)

Question. The point \( (a^2, a+1) \) lies in the angle between the lines \( 3x - y + 1 = 0 \) and \( x + 2y - 5 = 0 \) containing the origin if
(a) \( a \in (-3, 0) \cup \left(\frac{1}{3}, 1\right) \)
(b) \( a \in \left(-3, \frac{1}{3}\right) \)
(c) \( a \in (-\infty, -3) \cup \left(\frac{1}{3}, 1\right) \)
(d) \( a \in \left(\frac{1}{3}, \infty\right) \)
Answer: (a) \( a \in (-3, 0) \cup \left(\frac{1}{3}, 1\right) \)

Question. On the portion of the straight line \( x + y = 4 \) which is intercepted between the axes, a square is constructed away from the origin, with the portion as one of its side. If ‘d’ denote the perpendicualr distance of a side of this squre from the origin, then maximum value of ‘d’ is
(a) \( 2\sqrt{2} \)
(b) \( 3\sqrt{2} \)
(c) \( 4\sqrt{2} \)
(d) \( 6\sqrt{2} \)
Answer: (d) \( 6\sqrt{2} \)

Question. The equation of line segment \( AB \) is \( y = x \). If \( A \) & \( B \) lie on same side of line mirror \( 2x - y = 1 \), then the equation of image of \( AB \) with respect to line mirror \( 2x - y = 1 \) is
(a) \( y = 7x - 5 \)
(b) \( y = 7x - 6 \)
(c) \( y = 3x - 7 \)
(d) \( y = 6x - 5 \)
Answer: (b) \( y = 7x - 6 \)

Question. A line passing through \( P(6, 4) \) meets the coordinates axes at \( A \) and \( B \) resepctively. If \( O \) is the origin, then locus of the centre of the circumcircle of triangle \( OAB \) is
(a) \( 3x^{-1} + y^{-1} = 1 \)
(b) \( x^{-1} + 2y^{-1} = 1 \)
(c) \( x^{-1} + y^{-1} = 1 \)
(d) \( 3x^{-1} + 2y^{-1} = 1 \)
Answer: (d) \( 3x^{-1} + 2y^{-1} = 1 \)

Question. Let \( ax + by + c = 0 \) be a variable straight line, where \( a, b \) and \( c \) are \( 1^{st}, 3^{rd} \) and \( 7^{th} \) terms of some increasing A.P. Then the variable straight line always passes through a fixed point which lies on
(a) \( x^2 + y^2 = 13 \)
(b) \( x^2 + y^2 = 5 \)
(c) \( y^2 = 9x \)
(d) \( 3x + 4y = 9 \)
Answer: (a) \( x^2 + y^2 = 13 \)

Question. Line \( L \) has intercepts \( a \) and \( b \) on the coordinate axes. When the axes are rotated through a fixed given angle keeping the origin fixed, the same line \( L \) has intercepts \( p \) and \( q \), then
(a) \( a^2 + b^2 = p^2 + q^2 \)
(b) \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{p^2} + \frac{1}{q^2} \)
(c) \( a^2 + p^2 = b^2 + q^2 \)
(d) \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{b^2} + \frac{1}{q^2} \)
Answer: (b) \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{p^2} + \frac{1}{q^2} \)

Question. Given the family of lines, \( a(2x + y + 4) + b(x - 2y - 3) = 0 \). Among the lines of the family, the number of lines situated at a distance of \( \sqrt{10} \) from the point \( M(2, -3) \) is
(a) 0
(b) 1
(c) 2
(d) \( \infty \)
Answer: (b) 1

Question. If the angle between the lines represented by \( 6x^2 + 5xy - 4y^2 + 7x + 13y - 3 = 0 \) is \( \tan^{-1}(m) \) and \( a^2 + b^2 - ab - a - b + 1 \leq 0 \) then \( 5a + 6b \) is equal to
(a) \( m \)
(b) \( \frac{1}{m} \)
(c) \( 2m \)
(d) \( \frac{1}{2m} \)
Answer: (c) \( 2m \)

Question. PQR is an equilateral triangle such that the vertices Q and R lie on the lines \( x + y = \sqrt{2} \) and \( x + y = 7\sqrt{2} \) respectively. If P lies between the two lines at a distance 4 from one of them then the length of side of equilateral triangle PQR is (in units)
(a) 8
(b) \( \frac{4\sqrt{7}}{\sqrt{3}} \)
(c) \( \frac{\sqrt{85}}{3} \)
(d) \( \frac{4\sqrt{5}}{\sqrt{3}} \)
Answer: (b) \( \frac{4\sqrt{7}}{\sqrt{3}} \)

Question. \( P(m, n) (m, n \in N) \) is any point in the inerior of the quadrilateral formed by the pair of lines \( xy = 0 \) and two lines \( 2x + y - 2 = 0 \) and \( 4x + 5y = 20 \) then the possible number of positions of the points P is
(a) 4
(b) 5
(c) 6
(d) 11
Answer: (c) 6

Question. If \( d_1, d_2 \) denotes the lengths of the perpendiculars from the point (2, 3) on the lines given by \( 15x^2 + 31xy + 14y^2 = 0 \). If \( d_1 > d_2 \) then \( d_1^2 - d_2^2 + \frac{1}{74} + \frac{1}{13} = \)
(a) - 2
(b) 0
(c) 2
(d) 3
Answer: (c) 2

Question. Two of the straight lines given by \( 3x^3 + 3x^2y - 3xy^2 + py^3 = 0 \) are at right angles. The equation of line passing through (1, 1) and perpendicular to \( y = px \) is
(a) \( x + 3y - 4 = 0 \)
(b) \( 3x - y - 2 = 0 \)
(c) \( 3x + y - 4 = 0 \)
(d) \( x - 3y + 2 = 0 \)
Answer: (d) \( x - 3y + 2 = 0 \)

Question. ABCD is a square whose vertices \( A(0,0), B(2,0), C(2,2), D(0, 2) \). This square is rotated in the xy-plane with an angle of \( 30^\circ \) in anticlockwise direction about an axis passing through the vertex A, the equation of the diagonal BD of this rotated square is
(a) \( \sqrt{3}x + (1 - \sqrt{3})y = \sqrt{3} \)
(b) \( (1 + \sqrt{3})x - (1 - \sqrt{2})y = 2 \)
(c) \( (2 - \sqrt{3})x + y = 2(\sqrt{3} - 1) \)
(d) \( \sqrt{3}x + (1 - \sqrt{2})y = \sqrt{3} - 1 \)
Answer: (c) \( (2 - \sqrt{3})x + y = 2(\sqrt{3} - 1) \)

Question. The equations to a pair of opposite sides of parallelogram are \( x^2 - 5x + 6 = 0 \) and \( y^2 - 6y + 5 = 0 \), the equations to its diagonals are
(a) \( x + 4y = 13, y = 4x - 7 \)
(b) \( 4x + y = 13, 4y = x - 7 \)
(c) \( 4x + y = 13, y = 4x - 7 \)
(d) \( y - 4x = 13, y + 4x = 7 \)
Answer: (c) \( 4x + y = 13, y = 4x - 7 \)

Question. Two sides of a triangle having the joint equation \( (x - 3y + 1)(x + y - 2) = 0 \) the third side which is variable always passes through the point \( (-5, -1) \), then possible values of slope of the third side such that origin is an interior point of triangle are..
(a) \( \left(-1, \frac{-1}{5}\right) \)
(b) \( \left(\frac{-1}{5}, -1\right) \)
(c) \( (0, 2) \)
(d) \( \mathbb{R} \)
Answer: (a) \( \left(-1, \frac{-1}{5}\right) \)

Question. Through the point \( P(3,4) \) a pair of perpendicular lines are drawn which meet x-axis at the points A and B. The locus of incentre of triangle PAB is
(a) \( x^2 - y^2 - 6x - 8y + 25 = 0 \)
(b) \( x^2 + y^2 - 6x - 8y + 25 = 0 \)
(c) \( x^2 - y^2 + 6x + 8y + 25 = 0 \)
(d) \( x^2 + y^2 + 6x + 8y + 25 = 0 \)
Answer: (a) \( x^2 - y^2 - 6x - 8y + 25 = 0 \)

Question. Two parallel lines lying in the same quadrant make intercepts \( a \) and \( b \) on x, y axes respectively between them, then the distance between the lines is
(a) \( \frac{ab}{\sqrt{a^2 + b^2}} \)
(b) \( \sqrt{a^2 + b^2} \)
(c) \( \frac{1}{\sqrt{a^2 + b^2}} \)
(d) \( \frac{1}{a^2} + \frac{1}{b^2} \)
Answer: (a) \( \frac{ab}{\sqrt{a^2 + b^2}} \)

Question. The four sides of a quadrilateral are given by \( xy(x - 2)(y - 3) = 0 \). The equation of the line parallel to \( x - 4y = 0 \) that divides the quadrilateral into two equal areas is
(a) \( x - 4y + 7 = 0 \)
(b) \( x - 4y + 5 = 0 \)
(c) \( x - 4y + 11 = 0 \)
(d) \( x - 4y + 3 = 0 \)
Answer: (b) \( x - 4y + 5 = 0 \)

Question. Each side of triangle ABC is divided into three equal parts as shown in the figure ratio of area of hexagon PQRSTU to area of triangle ABC is
(a) 5/9
(b) 2/3
(c) 1/2
(d) 3/4
Answer: (b) 2/3

Question. A line through \( P(3, 4) \) cuts the lines \( x = 6 \) and \( y = 8 \) at \( L \) and \( M \) respectively. \( Q \) is a variable point on the line such that \( \frac{1}{PQ} = \frac{1}{PL} + \frac{1}{PM} \) then the locus of \( Q \) is
(a) \( 4x + 3y - 36 = 0 \)
(b) \( x^2 + y^2 = 36 \)
(c) \( 3x - 4y - 36 = 0 \)
(d) \( 4x^2 - 9y^2 = 36 \)
Answer: (a) \( 4x + 3y - 36 = 0 \)

Question. ABCDEF is a regular hexagon in anticlockwise sense and \( A(2, 0), B(4, 0) \) then the coordinates of \( C \) are
(a) \( \left(\frac{9}{2}, \sqrt{3}\right) \)
(b) \( (5, \sqrt{3}) \)
(c) \( \left(\frac{5}{2}, \frac{\sqrt{3}}{2}\right) \)
(d) \( \left(2, \frac{\sqrt{3}}{2}\right) \)
Answer: (b) \( (5, \sqrt{3}) \)

Question. Lines \( (1 + \lambda)x + (4 - \lambda)y + (2 + \lambda) = 0 \) and \( (4 - \lambda)x - (1 + \lambda)y + (6 - 3\lambda) = 0 \) are concurrent at points A and B respectively and intersect at C then locus of centroid of \( \Delta ABC \) is (\( \lambda \) is parameter)
(a) \( \left(x + \frac{3}{2}\right)^2 + \left(y + \frac{7}{10}\right)^2 = \frac{17}{50} \)
(b) \( \left(x - \frac{3}{2}\right)^2 + \left(y - \frac{7}{10}\right)^2 = \frac{17}{50} \)
(c) \( \left(x - \frac{3}{2}\right)^2 + \left(y + \frac{7}{10}\right)^2 = \frac{17}{450} \)
(d) \( \left(x + \frac{3}{2}\right)^2 + \left(y + \frac{7}{10}\right)^2 = \frac{17}{450} \)
Answer: (d) \( \left(x + \frac{3}{2}\right)^2 + \left(y + \frac{7}{10}\right)^2 = \frac{17}{450} \)

Question. The line \( x + y = 1 \) meets x-axis at \( A \) and y-axis at \( B \). \( P \) is the midpoint of \( AB \). \( P_1 \) is the foot of the perpendicular from \( P \) to \( OA \); \( M_1 \) is that from \( P_1 \) to \( OP \); \( P_2 \) is that from \( M_1 \) to \( OA \) and so on. If \( P_n \) denotes the nth foot of the perpendicular on \( OA \) from \( M_{n-1} \) then \( OP_n \)
(a) 1/2
(b) \( 1/2^n \)
(c) \( 1/2^{n/2} \)
(d) \( 1/\sqrt{2} \)
Answer: (b) \( 1/2^n \)

Question. A ray of light leaves the point (3, 4) reflects off the y-axis towards x-axis and again after reflecting from x-axis finally arrives at the point (8, 2) then the abscissa of point where the reflected ray meets x-axis is
(a) \( \frac{9}{2} \)
(b) \( \frac{13}{3} \)
(c) \( \frac{14}{3} \)
(d) \( \frac{16}{3} \)
Answer: (b) \( \frac{13}{3} \)

Question. If the lengths of the medians through acute angles of a right angled triangle are 3 and 4 then the area of the triangle is (in sq.units)
(a) \( \frac{4\sqrt{11}}{3} \)
(b) \( \frac{2\sqrt{11}}{3} \)
(c) \( 2\sqrt{11} \)
(d) \( 3\sqrt{11} \)
Answer: (a) \( \frac{4\sqrt{11}}{3} \)

Question. A line is drawn through the point (- 4, 5) such that the distance of the point (-3, 2) from the line is \( d \) then the maximum value of \( d \)
(a) 0
(b) \( \sqrt{10} \)
(c) \( \sqrt{\frac{5}{2}} \)
(d) \( \sqrt{5} \)
Answer: (b) \( \sqrt{10} \)

Question. A lattice point in a plane is a point for which both coordinates are integers. The number of lattice points inside the triangle whose sides are \( x = 0, y = 0 \) and \( 9x + 223y = 2007 \) is
(a) 198
(b) 173
(c) 99
(d) 888
Answer: (d) 888

Question. A point \( P(x, y) \) moves such that the sum of its distances from the lines \( 2x - y - 3 = 0 \) and \( x + 3y + 4 = 0 \) is 7. The area bounded by locus of \( P \) is (in sq.units)
(a) 70
(b) \( 70\sqrt{2} \)
(c) \( 35\sqrt{2} \)
(d) 140
Answer: (b) \( 70\sqrt{2} \)

Question. A triangle ABC right angled at C moves such that A and B always lie on the positive x and y- axes then locus of C is
(a) Straight Line
(b) Circle
(c) Parabola
(d) Ellipse
Answer: (a) Straight Line

Question. The slopes of sides of a triangle are -1, -2, 3. If the orthocentre of the triangle is the origin O; then the locus of its centroid is \( \frac{y}{x} = \)
(a) 2/3
(b) 2/5
(c) 2/7
(d) 2/9
Answer: (d) 2/9

Question. Let \( \alpha, \beta \) be the roots of \( ax^2 + 2hx + b = 0 \) and \( \gamma, \delta \) be the roots of \( a_1x^2 + 2h_1x + b_1 = 0 \). Consider the points \( A(\alpha, 0), B(\beta, 0), C(\gamma, 0), D(\delta, 0) \). If the sum of ratios in which C and D divides AB is zero then \( ab_1 + a_1b = \)
(a) 0
(b) \( hh_1 \)
(c) \( 2hh_1 \)
(d) -1
Answer: (c) \( 2hh_1 \)

Question. Let PQR be right angled isoscles triangle right angled at \( P(2, 1) \). If the equation of the line QR is \( 2x + y = 3 \), then the equation representing the pair of lines PQ and PR is
(a) \( 3x^2 - 3y^2 + 8xy + 20x + 10y + 25 = 0 \)
(b) \( 3x^2 - 3y^2 + 8xy - 20x - 10y + 25 = 0 \)
(c) \( 3x^2 - 3y^2 + 8xy + 20x + 15y + 25 = 0 \)
(d) \( 3x^2 - 3y^2 + 8xy - 10x - 15y - 25 = 0 \)
Answer: (b) \( 3x^2 - 3y^2 + 8xy - 20x - 10y + 25 = 0 \)

Question. Let PS be the median of the triangle with vertices \( P(2, 2), Q(6, -1) \) and \( R(7, 3) \). The equation of the line passing through \( (1, -1) \) and parallel to PS is
(a) \( 2x - 9y - 7 = 0 \)
(b) \( 2x - 9y - 11 = 0 \)
(c) \( 2x + 9y - 11 = 0 \)
(d) \( 2x + 9y + 7 = 0 \)
Answer: (d) \( 2x + 9y + 7 = 0 \)