Class 11 Mathematics Straight Lines MCQs Set 08

Question. The separate equations of the lines represented by the equation \( (2x+3y)^2 - (x-2y)^2 = 0 \) are
(a) 3x+y=0; x+5y=0
(b) 3x-y=0; x-5y=0
(c) x+3y=0; 5x-y=0
(d) x+y = 0; x – 5y = 0
Answer: (a) 3x+y=0; x+5y=0

Question. The equation \( 3x^2 + 10xy - 8y^2 = 0 \) represents
(a) real and distinct lines
(b) coincident lines
(c) imaginary lines
(d) parallel lines
Answer: (a) real and distinct lines

Question. If \( 6x^2 - 5xy + y^2 = 0 \) represents a pair of lines then
I: \( m_1 + m_2 = 5 \) II: \( |m_1 - m_2| = 1 \)
Which of the above statements are correct
(a) only I
(b) only II
(c) both I and II
(d) neither I nor II
Answer: (c) both I and II

Question. If s and p are respectively the sum and the product of the slope of the lines \( 3x^2 - 2xy - 15y^2 = 0 \), then s:p=
(a) 4:3
(b) 2:3
(c) 3:5
(d) 3:4
Answer: (b) 2:3

Question. The difference of the slopes of the lines represented by \( x^2(\tan^2 \theta + \cos^2 \theta) + 2xy \tan \theta + y^2 \sin^2 \theta = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The combined equation of the lines passing through the origin and having slopes 3 and -2 is
(a) \( 6x^2 - xy + y^2 = 0 \)
(b) \( x^2 + xy - 6y^2 = 0 \)
(c) \( 6x^2 + xy - y^2 = 0 \)
(d) \( x^2 - xy + 6y^2 = 0 \)
Answer: (c) \( 6x^2 + xy - y^2 = 0 \)

Question. If the slopes of the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) are in the ratio 3 : 2, then
(a) \( 25ab = 24h^2 \)
(b) \( 8h^2 = 9ab \)
(c) \( 16h^2 = 25ab \)
(d) \( h^2 = ab \)
Answer: (a) \( 25ab = 24h^2 \)

Question. If the sum of the slopes of the lines given by \( x^2 - cxy - 7y^2 = 0 \) is four times their product, then c has the value
(a) 2
(b) -1
(c) 1
(d) -2
Answer: (a) 2

Question. If the acute angle between the pair of lines \( 2x^2 + 5xy + 3y^2 = 0 \) is \( \tan^{-1} k \) then k =
(a) 1/5
(b) 1
(c) 7/5
(d) 7
Answer: (a) 1/5

Question. If the angle \( 2\theta \) is acute, then the acute angle between the pair of straight lines \( x^2(\cos \theta - \sin \theta) + 2xy \cos \theta + y^2(\cos \theta + \sin \theta) = 0 \) is
(a) \( 2\theta \)
(b) \( \frac{\theta}{2} \)
(c) \( \frac{\theta}{3} \)
(d) \( \theta \)
Answer: (d) \( \theta \)

Question. If the pair of lines represented by \( (x^2 + y^2) \tan^2 \alpha = (x - y \tan \alpha)^2 \) are perpendicular to each other, then \( \alpha = \)
(a) \( \pi/6 \)
(b) \( \pi/3 \)
(c) \( \pi/8 \)
(d) \( \pi/4 \)
Answer: (d) \( \pi/4 \)

Question. If \( \theta \) is the acute angle between the lines \( 6x^2 + 11xy + 3y^2 = 0 \), then \( \tan \theta = \)
(a) 9/7
(b) 7/9
(c) 3/7
(d) 7/3
Answer: (b) 7/9

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

Question. The equation of the pair of the lines through (1, -1) and perpendicular to the pair of lines \( x^2 - xy - 2y^2 = 0 \) is
(a) \( 2x^2 - xy + y^2 + 5x + y + 2 = 0 \)
(b) \( 2x^2 - xy - y^2 - 5x - y + 2 = 0 \)
(c) \( 2x^2 - xy + y^2 - 5x - y - 2 = 0 \)
(d) \( 2x^2 - xy - y^2 + 5x + y - 2 = 0 \)
Answer: (b) \( 2x^2 - xy - y^2 - 5x - y + 2 = 0 \)

Question. The product of perpendiculars from ( 0,1 ) to the pair of lines \( 2x^2 - 5xy + y^2 = 0 \) is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{\sqrt{26}} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{\sqrt{13}} \)
Answer: (b) \( \frac{1}{\sqrt{26}} \)

Question. If the product of the perpendicular distances from (l,k) to the pair of lines \( x^2 - 4xy + y^2 = 0 \) is 3/2 units, then k =
(a) 4
(b) 5
(c) 6
(d) 8
Answer: (b) 5

Question. The area of the triangle formed by the lines \( x^2 + 4xy + y^2 = 0, x+y=1 \) is
(a) \( \sqrt{3} \)
(b) 2
(c) 1
(d) \( \sqrt{3}/2 \)
Answer: (d) \( \sqrt{3}/2 \)

Question. The area (in square units) of the triangle formed by \( x+y+1=0 \) and the pair of straight lines \( x^2 - 3xy + 2y^2 = 0 \) is:
(a) 7/12
(b) 5/12
(c) 1/12
(d) 1/6
Answer: (c) 1/12

Question. Match the following :
List - I (sides of the triangle)
A. \( x^2 - 4xy + y^2 = 0 \) and \( x+y=1 \)
B. \( xy = 0 \) and \( x+y=1 \)
C. \( x^2 + 4xy + y^2 = 0 \) and \( x+y=1 \)

List - II (Area of the triangle)
1. \( \frac{1}{2} \) sq.u
2. \( \frac{1}{2\sqrt{3}} \) sq.u
3. \( \frac{\sqrt{3}}{2} \) sq.u
4. \( \frac{1}{\sqrt{3}} \) sq.u
(a) A-2, B-1, C-4
(b) A-2, B-1, C-3
(c) A-1, B-2, C-4
(d) A-1, B-2, C-3
Answer: (b) A-2, B-1, C-3

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

Question. The area of the equilateral triangle formed by the lines \( x^2 - 3y^2 = 0 \) and the line \( x - 3a = 0 \), in sq.units is
(a) \( \sqrt{3}a^2 \)
(b) \( 2\sqrt{3}a^2 \)
(c) \( 3\sqrt{3}a^2 \)
(d) \( 4\sqrt{3}a^2 \)
Answer: (c) \( 3\sqrt{3}a^2 \)

Question. If a, h, b are in A.P. then the triangle area formed by the pair of lines \( ax^2 + 2hxy + by^2 = 0 \) and the line \( x - y = -2 \) in square units is
(a) \( | \frac{a+b}{a-b} | \)
(b) \( | \frac{a^2+b^2}{a-b} | \)
(c) \( | \frac{a-b}{a+b} | \)
(d) \( | \frac{a^2+b^2}{a+b} | \)
Answer: (c) \( | \frac{a-b}{a+b} | \)

Question. The equation of the bisectors of the angle between the two straight lines \( x^2 - xy - 6y^2 = 0 \) is
(a) \( x^2 + 14xy - y^2 = 0 \)
(b) \( x^2 + 14xy + y^2 = 0 \)
(c) \( x^2 - 14xy - y^2 = 0 \)
(d) \( x^2 - 14xy + y^2 = 0 \)
Answer: (a) \( x^2 + 14xy - y^2 = 0 \)

Question. The pair of straight lines \( h(x^2 - y^2) + pxy = 0 \) bisects the angle between the pair of lines \( ax^2 + 2hxy + by^2 = 0 \) then the value of p is
(a) a - b
(b) b - a
(c) a + b
(d) -a - b
Answer: (b) b - a

Question. If one of the lines \( my^2 + (1-m^2)xy - mx^2 = 0 \) is a bisector of the angle between the lines \( xy = 0 \), then m is
(a) -1/2
(b) -2
(c) 1
(d) 2
Answer: (c) 1

Question. If one of the lines in the pair of straight lines given by \( x^2 + (2+k)xy - 4y^2 = 0 \) bisects the angle between the coordinate axes, then \( k \in \)
(a) {-1,-5}
(b) {-1, 5}
(c) {1, -5}
(d) {1, 5}
Answer: (c) {1, -5}

Question. If the pairs of lines \( x^2 - pxy - 3y^2 = 0 \) and \( x^2 - qxy - 5y^2 = 0 \) are such that each pair bisects the angle between the other pair, then pq equals to
(a) -1
(b) -7
(c) -9
(d) -15
Answer: (d) -15

Question. The equation of the pair of bisectors of the angles between the pair of lines \( x^2 + axy - y^2 = 0 \) is \( x^2 + bxy - y^2 = 0 \). Then
(a) ab = 1
(b) ab + 1 = 0
(c) ab = 2
(d) ab + 2 = 0
Answer: (b) ab + 1 = 0

Question. If \( 2x+3y=7 \) makes equal angles with \( 9x^2 + 12xy + ky^2 = 0 \) then k =
(a) 3
(b) 7
(c) 14
(d) 21
Answer: (c) 14

Question. The pair of lines \( 2x^2+3xy+5y^2=0 \); \( 4x^2+21xy+25y^2=0 \) are
(a) mutually perpendicular
(b) equally inclined to the axes
(c) equally inclined to each other
(d) Each pair bisects the angle between the other
Answer: (c) equally inclined to each other

Question. \( 2x^2 + 5xy + 3y^2 + 6x + 7y + 4 = 0 \) represents two lines \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \) then \( m_1 + m_2 \) and \( m_1 m_2 \) are
(a) -5/3, -2/3
(b) -5/3, 2/3
(c) 5/3, -2/3
(d) 5/3, 2/3
Answer: (b) -5/3, 2/3

Question. If \( 4xy + 2x + 2fy + 3 = 0 \) represents a pair of lines then f =
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 3

Question. If the equation \( 2xy + gx + fy + 5 = 0 \) represents a pair of lines, then fg =
(a) 1
(b) 10
(c) 5
(d) 0
Answer: (b) 10

Question. The value of \( \lambda \) such that \( \lambda x^2 - 10xy + 12y^2 + 5x - 16y - 3 = 0 \) represents a pair of straight lines is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (c) 2

Question. The value of \( \lambda \) with \( |\lambda| < 16 \) such that \( 2x^2 - 10xy + 12y^2 + 5x + \lambda y - 3 = 0 \) represents a pair of straight lines is
(a) -10
(b) -9
(c) 10
(d) 9
Answer: (b) -9

Question. \( x^2 + k_1y^2 + 2k_2y = a^2 \) represents a pair of perpendicular lines if
(a) \( k_1 = 1, k_2 = a \)
(b) \( k_1 = 1, k_2 = -a \)
(c) \( k_1 = -1, k_2 = -a \)
(d) \( k_1 = -1, k_2 = a^2 \)
Answer: (c) \( k_1 = -1, k_2 = -a \)

Question. The angle between the pair of lines \( 2x^2 + 5xy + 2y^2 + 3x + 3y + 1 = 0 \) is
(a) \( \cos^{-1} \left( \frac{4}{5} \right) \)
(b) \( \tan^{-1} \left( \frac{4}{5} \right) \)
(c) 0
(d) \( \pi/2 \)
Answer: (a) \( \cos^{-1} \left( \frac{4}{5} \right) \)

Question. The angle between the pair of lines \( 2(x+2)^2 + 3(x+2)(y-2) - 2(y-2)^2 = 0 \) is
(a) \( \pi/4 \)
(b) \( \pi/3 \)
(c) \( \pi/6 \)
(d) \( \pi/2 \)
Answer: (d) \( \pi/2 \)

Question. The equation \( x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0 \) when \( \lambda \) is a real number, represents a pair of straight lines. If \( \theta \) is the angle between these lines, then \( \csc^2 \theta = \)
(a) 3
(b) 9
(c) 10
(d) 100
Answer: (c) 10

Question. The equation to the pair of lines through the origin perpendicular to the pair of lines \( 2x^2 + 5xy + 2y^2 + 10x + 5y = 0 \) is
(a) \( 2x^2 + 5xy + 2y^2 = 0 \)
(b) \( 2x^2 - 5xy + 2y^2 = 0 \)
(c) \( 2x^2 - 5xy - 2y^2 = 0 \)
(d) \( x^2 - 5xy + y^2 = 0 \)
Answer: (b) \( 2x^2 - 5xy + 2y^2 = 0 \)