Class 11 Mathematics Straight Lines MCQs Set 06

Question. The equation \( ax^2 + 2\sqrt{ab}xy + by^2 = 0 \) represents a pair of lines which are
(a) perpendicular
(b) coincident
(c) imaginary
(d) parallel
Answer: (b) coincident

Question. If one of the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) is the Y-axis then the equation of the other line is
(a) \( ax + 2hy = 0 \)
(b) \( 2hx + by = 0 \)
(c) \( ax + by = 0 \)
(d) \( hx + by = 0 \)
Answer: (a) \( ax + 2hy = 0 \)

Question. \( y = mx \) is one of the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) if \( a + 2hm + bm^2 = \)
(a) -1
(b) 1
(c) 0
(d) 2
Answer: (c) 0

Question. If the pair of straight lines \( Ax^2 + 2Hxy + By^2 = 0 \) (\( H^2 > AB \)) forms an equilateral triangle with the line \( ax + by + c = 0 \) then \( (A + 3B)(3A + B) = \)
(a) \( H^2 \)
(b) \( -H^2 \)
(c) \( 2H^2 \)
(d) \( 4H^2 \)
Answer: (d) \( 4H^2 \)

Question. The equation of the line common to the pair of lines \( (p^2 - q^2)x^2 + (q^2 - r^2)xy + (r^2 - p^2)y^2 = 0 \) and \( (l - m)x^2 + (m - n)xy + (n - l)y^2 = 0 \) is
(a) x + y = 0
(b) x - y = 0
(c) x + y = pqr
(d) x - y = pqr
Answer: (b) x - y = 0

Question. The equation of the line common to the pair of lines \( m^2x^2 - (m^2 + 1)xy + y^2 = 0 \) and \( mx^2 - (m + 1)xy + y^2 = 0 \) is
(a) mx - y = 0
(b) x + y = 0
(c) x - y = 0
(d) x + y = m
Answer: (c) x - y = 0

Question. If \( x^2 + axy + bcy^2 = 0 \) and \( x^2 + bxy + cay^2 = 0 \) have a common line then the equation of that line is
(a) y = cx
(b) x = cy
(c) x = ay
(d) x = by
Answer: (b) x = cy

Question. The locus of the points equidistant from the pair of lines \( x^2 \cos^2 \theta - 2xy \sin^2 \theta - y^2 \sin^2 \theta = 0 \) is
(a) \( x^2 - y^2 + 2xy \text{ cosec } 2\theta = 0 \)
(b) \( x^2 + y^2 + 2xy \text{ cosec } 2\theta = 0 \)
(c) \( x^2 - y^2 + 2xy \sec^2 \theta = 0 \)
(d) \( x^2 + y^2 + 2xy \sec^2 \theta = 0 \)
Answer: (a) \( x^2 - y^2 + 2xy \text{ cosec } 2\theta = 0 \)

Question. The separate equations of the angular bisectors of the pair of lines \( (ax + by)^2 - (bx - ay)^2 = 0 \) are
(a) ax - by = 0, bx - ay = 0
(b) ax + by = 0, bx - ay = 0
(c) ax + by = 0, bx + ay = 0
(d) (x + y) = 0, (x - y) = 0
Answer: (b) ax + by = 0, bx - ay = 0

Question. The slope of angular bisectors of pair of lines \( (ax + by)^2 = c(bx - ay)^2 \), (\( c > 0 \)) are
(a) \( -\frac{a}{b}, \frac{b}{a} \)
(b) \( \frac{a}{b}, -\frac{b}{a} \)
(c) \( \frac{a}{c}, -\frac{c}{a} \)
(d) \( \frac{c}{a}, -\frac{a}{c} \)
Answer: (a) \( -\frac{a}{b}, \frac{b}{a} \)

Question. The equation to the image of the pair of lines \( ax^2 + 2hxy + by^2 = 0 \) where \( h^2 > ab \) with respect to \( y = 0 \) is
(a) \( bx^2 + 2hxy + ay^2 = 0 \)
(b) \( bx^2 - 2hxy + ay^2 = 0 \)
(c) \( ax^2 - 2hxy + by^2 = 0 \)
(d) \( ax^2 + 2hxy + by^2 = 0 \)
Answer: (c) \( ax^2 - 2hxy + by^2 = 0 \)

Question. Point of intersection of pair of lines \( a(x - \alpha)^2 + 2h(x - \alpha)(y - \beta) + b(y - \beta)^2 = 0 \) is
(a) (0, 0)
(b) \( (\alpha, \beta) \)
(c) \( (-\alpha, -\beta) \)
(d) \( (-\alpha, \beta) \)
Answer: (b) \( (\alpha, \beta) \)

Question. If the pair of lines given by \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) intersect on X-axis then
(a) b, f, c are in A.P
(b) a, f, c are in G.P
(c) a, g, c are in G.P
(d) a, g, c are in A.P
Answer: (c) a, g, c are in G.P

Question. The product of the perpendiculars from (f, -g) to the pair of st.lines \( hxy + gx + fy + c = 0 \) is
(a) \( |fg| \)
(b) \( \left| \frac{fg - c}{h} \right| \)
(c) \( |fgh - c| \)
(d) \( \left| \frac{fgh - c}{h} \right| \)
Answer: (d) \( \left| \frac{fgh - c}{h} \right| \)

Question. The rectangle formed by the pair of lines \( 2hxy + 2gx + 2fy + c = 0 \) with the coordinate axes has the area equal to
(a) \( \left| \frac{fg}{h} \right| \)
(b) \( \left| \frac{gh}{f^2} \right| \)
(c) \( \left| \frac{hf}{g^2} \right| \)
(d) \( \frac{|fg|}{h} \)
Answer: (a) \( \left| \frac{fg}{h} \right| \)

Question. If the pair of lines \( 2hxy + 2gx + 2fy + c = 0 \) and the coordinate axes form a rectangle, then the equations of its diagonals are
(a) \( 2gx + 2fy + c = 0, gx - fy = 0 \)
(b) \( 2gx + 2fy - c = 0, gx + fy = 0 \)
(c) \( gx + fy - c = 0, gx - fy = 0 \)
(d) \( gx + fy = 0, gx - fy = 0 \)
Answer: (a) \( 2gx + 2fy + c = 0, gx - fy = 0 \)

Question. The lines \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) intersect x-axis in A, B and y-axis in C, D respectively. Then the combined equation of \( \overline{AB} \) and \( \overline{CD} \) is
(a) \( xy = 0 \)
(b) \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \)
(c) \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c + \frac{4gf}{c} xy = 0 \)
(d) \( ax^2 - 2hxy + by^2 + 2gx + 2fy + c + \frac{4gf}{c} xy = 0 \)
Answer: (a) \( xy = 0 \)

Question. The locus represented by the equation \( (x - y + c)^2 + (x + y - c)^2 = 0 \) is
(a) A line parallel to x-axis
(b) A point
(c) Pair of lines
(d) Line parallel to y-axis
Answer: (b) A point

Question. The square of the distance of the point of intersection of the lines \( 6x^2 – 5xy – 6y^2 + x + 5y – 1 = 0 \) from the origin is
(a) 74/169
(b) 85/169
(c) 74/185
(d) 2/13
Answer: (d) 2/13

Question. If the pair of straight lines \( xy - x - y + 1 = 0 \) and the line \( ax + 2y - 3 = 0 \) are concurrent then a =
(a) -1
(b) 3
(c) 1
(d) 0
Answer: (c) 1

Question. If the lines represented by \( ax^2 + 4xy + y^2 + 8x + 2fy + c = 0 \) intersect on Y-axis, then (f, c) =
(a) (2, 4)
(b) (4, 2)
(c) (-2, -4)
(d) (-4, -2)
Answer: (a) (2, 4)

Question. The intercept made by the pair of lines \( 6x^2 - 7xy - 3y^2 - 24x – 3y + 18 = 0 \) on the X-axis is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (a) 2

Question. Arrange A,B,C in descending order:
A: Intercept made by the pair of lines \( 6x^2 - 7xy - 3y^2 - 24x - 3y + 18 = 0 \) on X-axis
B: Intercept made by the pair of lines \( 2x^2 + 4xy - 6y^2 + 3x + y + 1 = 0 \) on Y-axis
C: The distance between the parallel lines \( x^2 + 2\sqrt{3}xy + 3y^2 - 3x - 3\sqrt{3}y - 4 = 0 \)
(a) A, B, C
(b) C, A, B
(c) B, C, A
(d) C, B, A
Answer: (b) C, A, B

Question. The figure formed by the four lines \( 3x^2 + 10xy + 3y^2 = 0 \) and \( 3x^2 + 10xy + 3y^2 - 28x - 28y + 49 = 0 \)
(a) parallelogram
(b) rhombus
(c) rectangle
(d) square
Answer: (b) rhombus

Question. If the adjacent sides of a parallelogram are \( 2x^2 - 5xy + 3y^2 = 0 \) and one diagonal is \( x+y+2=0 \) then the other diagonal is
(a) 9x-11y=0
(b) 9x+11y=0
(c) 11x-9y=0
(d) 11x+9y=0
Answer: (a) 9x-11y=0

Question. The area of the square formed by the lines \( 6x^2 - 5xy - 6y^2 = 0 \) and \( 6x^2 - 5xy - 6y^2 + x + 5y - 1 = 0 \) in sq. units is
(a) 1 / \( \sqrt{3} \)
(b) 4 / \( \sqrt{13} \)
(c) \( \sqrt{13} \)
(d) 1/13
Answer: (d) 1/13

Question. The equation to the pair of lines joining the origin to the points of intersection of \( y = x+3 \) and \( 2x^2 + 2y^2 = 1 \) is
(a) \( 12(x^2 + y^2) = (x-y)^2 \)
(b) \( 6(x^2 + y^2) = (x-y)^2 \)
(c) \( 18(x^2 + y^2) = (x-y)^2 \)
(d) \( 2(x^2 + y^2) = (x-y)^2 \)
Answer: (c) \( 18(x^2 + y^2) = (x-y)^2 \)

Question. A pair of perpendicular straight lines passes through the origin and also through the point of intersection of the curve \( x^2 + y^2 = 4 \) with \( x + y = a \). The set containing the value of ‘a’ is
(a) {-2,2}
(b) {-3,3}
(c) {-4,4}
(d) {-5,5}
Answer: (a) {-2,2}

Question. The angle between the pair of straight lines formed by joining the points of intersection of \( x^2 + y^2 = 4 \) and \( y = x + c\sqrt{3} \) to the origin is a right angle. Then \( c^2 \) is equal to 
(a) 20
(b) 13
(c) 1/5
(d) 5
Answer: (a) 20

Question. The triangle formed by the pair of lines \( 3x^2 + 48xy + 23y^2 = 0 \) and the line \( 3x - 2y + 4 = 0 \) is
(a) Equilateral
(b) Isosceles
(c) Right angled
(d) Scalene
Answer: (a) Equilateral

Question. If the angle between the pair of lines \( 2x^2 + \lambda xy + 3y^2 + 8x + 14y + 8 = 0 \) is \( \frac{\pi}{4} \), then the value of \( \lambda \) is
(a) \( \pm 6 \)
(b) \( \pm 7 \)
(c) \( \pm 5 \)
(d) \( \pm 1 \)
Answer: (b) \( \pm 7 \)

Question. The equation to the pair of straight lines passing through (2, 1) and perpendicular to the pair of lines \( 4xy + 2x + 6y + 3 = 0 \) is
(a) \( xy + x + 2y + 2 = 0 \)
(b) \( xy + x + 2y - 2 = 0 \)
(c) \( xy + x - 2y - 2 = 0 \)
(d) \( xy - x - 2y + 2 = 0 \)
Answer: (d) \( xy - x - 2y + 2 = 0 \)

Question. If the equation \( 2x^2 + 5xy + 3y^2 + 6x + 7y + 4 = 0 \) represents a pair of lines, then the equation of pair of lines parallel to them and passing through the point (0,1) is
(a) \( 2x^2 + 5xy + 3y^2 - 5x + 6y = 0 \)
(b) \( 2x^2 + 5xy + 3y^2 + 5x - 6y + 3 = 0 \)
(c) \( 2x^2 + 5xy + 3y^2 - 5x - 6y + 3 = 0 \)
(d) \( 2x^2 + 5xy + 3y^2 + 5x + 6y + 3 = 0 \)
Answer: (c) \( 2x^2 + 5xy + 3y^2 - 5x - 6y + 3 = 0 \)

Question. If the distance between the pair of parallel lines \( x^2 + 2xy + y^2 - 8ax - 8ay - 9a^2 = 0 \) is \( 25\sqrt{2} \) then a =
(a) 1
(b) 2
(c) 3
(d) 5
Answer: (d) 5

Question. The condition for the equation \( x^2 + (\lambda + \mu)xy + \lambda\mu y^2 + x + \mu y = 0 \) to represent pair of parallel lines and the distance between them are
(a) \( \lambda + \mu = 0, \sqrt{1 + \mu^2} \)
(b) \( \lambda + \mu = 0, \frac{1}{\sqrt{1 + \mu^2}} \)
(c) \( \lambda = \mu, \sqrt{1 + \mu^2} \)
(d) \( \lambda = \mu, \frac{1}{\sqrt{1 + \mu^2}} \)
Answer: (d) \( \lambda = \mu, \frac{1}{\sqrt{1 + \mu^2}} \)

Question. The distance between the parallel lines given by \( (x + 7y)^2 + 4\sqrt{2}(x + 7y) - 42 = 0 \) is
(a) 4/5
(b) \( 4\sqrt{2} \)
(c) 2
(d) \( 10\sqrt{2} \)
Answer: (b) \( 4\sqrt{2} \)

Question. The product of perpendicular distances from the origin to the pair of straight lines \( 12x^2 + 25xy + 12y^2 + 10x + 11y + 2 = 0 \)
(a) 1/25
(b) 2/25
(c) 3/25
(d) 4/25
Answer: (b) 2/25

Question. The point of the intersection of the pair of lines \( x^2 + xy + 2y^2 - 3x + 2y + 4 = 0 \) is
(a) (1, 2)
(b) (-1, 2)
(c) (-2, 1)
(d) (2, -1)
Answer: (d) (2, -1)