Class 11 Mathematics Straight Lines MCQs Set 07

Question. The separate equations of the lines \( 6x^2 + 5xy - 6y^2 = 0 \) are
(a) \( 2x-3y=0; 3x+2y=0 \)
(b) \( 2x+3y=0; 3x-2y=0 \)
(c) \( 2x-3y=0; 3x-2y=0 \)
(d) \( 3x+2y=0; 2x+3y=0 \)
Answer: (b) \( 2x+3y=0; 3x-2y=0 \)

Question. The range of ‘a’ so that \( a^2x^2 + 2xy + 4y^2 = 0 \) represents distinct lines
(a) \( a > \frac{1}{2} \) or \( a < -\frac{1}{2} \)
(b) \( -\frac{1}{2} \leq a \leq \frac{1}{2} \)
(c) \( -\frac{1}{2} < a < \frac{1}{2} \)
(d) \( a \geq \frac{1}{2} \) or \( a \leq -\frac{1}{2} \)
Answer: (c) \( -\frac{1}{2} < a < \frac{1}{2} \)

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

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

Question. The combined equation to a pair of straight lines passing through the origin and inclined at an angles \( 30^\circ \) and \( 60^\circ \) respectively with X-axis is
(a) \( \sqrt{3}(x^2 + y^2) = 4xy \)
(b) \( 4(x^2 + y^2) = \sqrt{3}xy \)
(c) \( x^2 + \sqrt{3}y^2 - 2xy = 0 \)
(d) \( x^2 + 3y^2 - 2xy = 0 \)
Answer: (b) \( 4(x^2 + y^2) = \sqrt{3}xy \)

Question. If the slope of one line is twice the slope of the other in the pair of straight lines \( ax^2 + 2hxy + by^2 = 0 \) then \( 8h^2 = \)
(a) 7 ab
(b) -7ab
(c) 9 ab
(d) -9ab
Answer: (c) 9 ab

Question. The equation of the pair of lines passing through the origin whose sum and product of slopes are respectively the arithmetic mean and geometric mean of 4 and 9 is
(a) \( 12x^2 - 13xy + 2y^2 = 0 \) 
(b) \( 12x^2 + 13xy + 2y^2 = 0 \)
(c) \( 12x^2 - 15xy + 2y^2 = 0 \)
(d) \( 12x^2 + 15xy - 2y^2 = 0 \)
Answer: (a) \( 12x^2 - 13xy + 2y^2 = 0 \)

Question. Assertion A : If \( ax^2 + 2hxy + by^2 = 0 \) represents two straight lines whose one slope is thrice the other then \( 3h^2 = 4ab \)
Reason R: If \( ax^2 + 2hxy + by^2 = 0 \) represents two lines and whose slopes are m:n then \( \frac{(m + n)^2}{4mn} = \frac{ab}{h^2} \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (c) A is true but R is false

Question. If the sum of the slopes of the lines given by \( x^2 + cxy + y^2 = 0 \) is eight times their product, then c has the value
(a) 1
(b) -1
(c) -4
(d) -2
Answer: (c) -4

Question. Angle between the lines \( x^2 + 2xy \sec \alpha + y^2 = 0 \) is
(a) \( \frac{\pi}{2} \)
(b) \( \alpha \)
(c) \( 2\alpha \)
(d) \( \frac{\alpha}{2} \)
Answer: (b) \( \alpha \)

Question. The angle between the pair of lines \( y^2 \cos^2 \theta - xy \cos^2 \theta + x^2(\sin^2 \theta - 1) = 0 \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{2} \)
Answer: (d) \( \frac{\pi}{2} \)

Question. If the pair of lines given by \( (x^2 + y^2)\sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 \) are perpendicular to each other then \( \alpha = \)
(a) \( \pi / 2 \)
(b) 0
(c) \( \pi / 4 \)
(d) \( \pi / 3 \)
Answer: (c) \( \pi / 4 \)

Question. If \( \theta \) is the acute angle between the pair of lines \( x^2 + 3xy - 4y^2 = 0 \) then \( \sin \theta = \)
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{5}{\sqrt{34}} \)
(d) \( \frac{3}{\sqrt{34}} \)
Answer: (c) \( \frac{5}{\sqrt{34}} \)

Question. The equation to the pair of lines passing through the point \( (-2,3) \) and parallel to the pair of lines \( x^2 + 4xy + y^2 = 0 \) is
(a) \( x^2 - 4xy + y^2 - 8x + 2y - 11 = 0 \)
(b) \( x^2 + 4xy + y^2 - 8x + 2y - 11 = 0 \)
(c) \( x^2 + 4xy - y^2 - 8x + 2y - 11 = 0 \)
(d) \( x^2 - 4xy + y^2 - 8x - 2y - 11 = 0 \)
Answer: (b) \( x^2 + 4xy + y^2 - 8x + 2y - 11 = 0 \)

Question. The equation to the pair of lines passing through the origin and perpendicular to \( 5x^2 + 3xy = 0 \)
(a) \( 5xy + 3y^2 = 0 \)
(b) \( x^2 - 2y^2 = 0 \)
(c) \( 3xy - 5y^2 = 0 \)
(d) \( 3x^2 - 2xy = 0 \)
Answer: (c) \( 3xy - 5y^2 = 0 \)

Question. The product of the perpendiculars from (-1, 2) to the pair of lines \( 2x^2 - 5xy + 2y^2 = 0 \)
(a) 4
(b) 3
(c) 8
(d) 5/2
Answer: (a) 4

Question. If the product of perpendiculars from (k, k) to the pair of lines \( x^2 + 4xy + 3y^2 = 0 \) is \( 4 / \sqrt{5} \) then k is
(a) \( \pm 4 \)
(b) \( \pm 3 \)
(c) \( \pm 2 \)
(d) \( \pm 1 \)
Answer: (d) \( \pm 1 \)

Question. Area of the triangle formed by the lines \( 2x - y = 6 \) and \( 3x^2 - 4xy + y^2 = 0 \) is
(a) 16
(b) 25
(c) 36
(d) 49
Answer: (c) 36

Question. If the area of the triangle formed by the pair of lines \( 8x^2 - 6xy + y^2 = 0 \) and the line \( 2x + 3y = a \) is 7 then a=
(a) 14
(b) \( 14\sqrt{2} \)
(c) \( 28\sqrt{2} \)
(d) 28
Answer: (d) 28

Question. If the area of the triangle formed by the lines \( 3x^2 - 2xy - 8y^2 = 0 \) and the line \( 2x+y-k=0 \) is 5sq. units, then k =
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (a) 5

Question. The area of the equilateral triangle formed by the lines passing through the origin and the line \( 12x-5y+13=0 \), in sq.units is
(a) \( \sqrt{3} / 3 \)
(b) \( 2\sqrt{3} \)
(c) \( \sqrt{3} \)
(d) \( 1 / \sqrt{3} \)
Answer: (d) \( 1 / \sqrt{3} \)

Question. If the sides of a triangle are \( ax^2 + 2hxy + by^2 = 0 \) and \( y = x+c \), then its area is
(a) \( \frac{c^2 \sqrt{h^2 - ab}}{|a+b+2h|} \)
(b) \( \frac{c \sqrt{h^2 - ab}}{a+b+2h} \)
(c) \( \frac{\sqrt{h^2 - ab}}{|a+b+c|} \)
(d) \( \frac{\sqrt{h^2 - ab}}{|a+b+2h|} \)
Answer: (a) \( \frac{c^2 \sqrt{h^2 - ab}}{|a+b+2h|} \)

Question. The equation of the bisectors of the angle between the two straight lines \( 2x^2 - 3xy + y^2 = 0 \) is
(a) \( 3x^2 - 2xy + 3y^2 = 0 \)
(b) \( x^2 + xy - y^2 = 0 \)
(c) \( 3x^2 + 2xy + 3y^2 = 0 \)
(d) \( 3x^2 + 2xy - 3y^2 = 0 \)
Answer: (d) \( 3x^2 + 2xy - 3y^2 = 0 \)

Question. If the equation of the pair of bisectors of the angle between the pair of lines \( 3x^2 + xy + by^2 = 0 \) is \( x^2 - 14xy - y^2 = 0 \) then b =
(a) 4
(b) -4
(c) 8
(d) -8
Answer: (b) -4

Question. If the lines \( x^2 + (2+k)xy - 4y^2 = 0 \) are equally inclined to the coordinate axes, then k =
(a) -1
(b) -2
(c) -3
(d) -4
Answer: (b) -2

Question. If the pair of straight lines \( x^2 - pxy - y^2 = 0 \) and \( x^2 - qxy - y^2 = 0 \) be such that each pair bisects the angle between the other pair, then
(a) \( pq = -1 \)
(b) \( p + q = 3 \)
(c) \( p - q = -4 \)
(d) \( pq = 1 \)
Answer: (a) \( pq = -1 \)

Question. If one of the lines in the pair of straight lines given by \( 4x^2 + 6xy + ky^2 = 0 \) bisects the angle between the coordinate axes, then k belongs to
(a) {-2,-10}
(b) {-2,10}
(c) {-10,2}
(d) {2,10}
Answer: (c) {-10,2}

Question. If \( x^2 - y^2 = 0 \), \( lx + 2y = 1 \) form an isosceles triangle then l =
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (d) 0

Question. If the two pairs of lines \( 2x^2 + 6xy + y^2 = 0 \) and \( 4x^2 + 18xy + by^2 = 0 \) are equally inclined, then b =
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (c) 2

Question. Two lines \( 9x^2 + y^2 + 6xy - 4 = 0 \) are
(a) parallel and coincident
(b) coincident only
(c) parallel but not coincident
(d) perpendicular
Answer: (c) parallel but not coincident

Question. If the lines represented by \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) are written in the form \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \), then
(a) \( m_1 + m_2 = a / b, m_1m_2 = 2h / b \)
(b) \( m_1 + m_2 = -2h / b, m_1m_2 = a / b \)
(c) \( m_1 + m_2 = -2h / b, m_1m_2 = -b / a \)
(d) \( m_1 + m_2 = 2h / b, m_1m_2 = a / b \)
Answer: (b) \( m_1 + m_2 = -2h / b, m_1m_2 = a / b \)

Question. The value k such that \( 3x^2 + 11xy + 10y^2 + 7x + 13y + k = 0 \) represents a pair of straight lines is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. If \( kx^2 + 10xy + 3y^2 - 15x - 21y + 18 = 0 \) represents a pair of straight lines then k =
(a) 3
(b) 4
(c) -3
(d) 5
Answer: (a) 3

Question. If \( x^2 + 4xy + 4y^2 + 4x + cy + 3 \) can be written as a product of two linear factors then c =
(a) 2
(b) 3
(c) 8
(d) 4
Answer: (c) 8

Question. The condition that the equation \( ax^2 + by^2 + c(x + y) = 0 \) to represents a pair of straight lines is
(a) \( a + b = 0 \) or \( c = 0 \)
(b) \( a + b = 0 \) or \( c = 0 \), \( ab < 0 \)
(c) \( ab > 0 \), \( c = 0 \)
(d) \( a + b \neq 0, c \neq 0 \)
Answer: (b) \( a + b = 0 \) or \( c = 0 \), \( ab < 0 \)

Question. If \( x^2 + \alpha y^2 + 2\beta y = a^2 \) represents a pair of perpendicular lines, then \( \beta = \) 
(a) 2a
(b) 3a
(c) 4a
(d) a
Answer: (d) a

Question. Angle between the pair of lines \( 2x^2 - 7xy + 3y^2 + 3x + y - 2 = 0 \)
(a) \( \pi / 2 \)
(b) \( \pi / 3 \)
(c) \( \pi / 4 \)
(d) \( \pi / 6 \)
Answer: (c) \( \pi / 4 \)

Question. The equation \( x^2 - 5xy + py^2 + 3x - 8y + 2 = 0 \) represents a pair of straight lines. If \( \theta \) is the angle between them, then \( \sin \theta = \) 
(a) \( \frac{1}{\sqrt{50}} \)
(b) \( \frac{1}{7} \)
(c) \( \frac{1}{5} \)
(d) \( \frac{1}{\sqrt{10}} \)
Answer: (a) \( \frac{1}{\sqrt{50}} \)

Question. The acute angle between the lines \( (5x - 2y)^2 - 3(2x + 5y)^2 = 0 \) is
(a) \( \pi / 6 \)
(b) \( \pi / 4 \)
(c) \( \pi / 3 \)
(d) \( \pi / 2 \)
Answer: (c) \( \pi / 3 \)

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