JEE Mathematics Pair of Straight Lines MCQs

Choose the most appropriate option (a, b, c or d).

Question. The equation \( 4x^2 + mxy - 3y^2 = 0 \) represents a pair of real and distinct lines if
(a) \( m \in R \)
(b) \( m \in (3, 4) \)
(c) \( m \in (-3, 4) \)
(d) \( m > 4 \)
Answer: (a) \( m \in R \)

Question. If \( 2x^2 + 3xy + my^2 = 0 \) represents two real and mutually perpendicular lines then m is
(a) any negative real number
(b) any positive real number
(c) -2
(d) none of the options
Answer: (c) -2

Question. The equation \( kx^2 + 4xy + 5y^2 = 0 \) represents two lines inclined at an angle \( \pi \) if k is
(a) \( \frac{5}{4} \)
(b) \( \frac{4}{5} \)
(c) \( -\frac{4}{5} \)
(d) none of the options
Answer: (b) \( \frac{4}{5} \)

Question. The equation \( x^3 + y^3 = 0 \) represents
(a) three real straight lines
(b) three points
(c) the combined equation of a straight line and a circle
(d) none of the options
Answer: (d) none of the options

Question. The angle between the pair of lines \( y^2 - 2xy \csc \theta + x^2 = 0, 0 \le \theta \le \frac{\pi}{2} \), is
(a) \( \frac{\pi}{2} \)
(b) \( \theta \)
(c) \( \frac{\pi}{2} - \theta \)
(d) none of the options
Answer: (c) \( \frac{\pi}{2} - \theta \)

Question. If the slope of one line is double the slope of another line and the combined equation of the pair of lines is \( \frac{x^2}{a} + \frac{2xy}{h} + \frac{y^2}{b} = 0 \) then \( ab : h^2 \) is
(a) 9 : 8
(b) 3 : 2
(c) 8 : 3
(d) none of the options
Answer: (a) 9 : 8

Question. The triangle formed by the lines whose combined equation is \( (y^2 - 4xy - x^2)(x + y - 1) = 0 \) is
(a) equilateral
(b) right angled
(c) isosceles
(d) obtuse angled
Answer: (b) right angled

Question. The three lines whose combined equation is \( y^3 - 4x^2y = 0 \) form a triangle which is
(a) isosceles
(b) equilateral
(c) right angled
(d) none of the options
Answer: (d) none of the options

Question. The combined equation of the lines \( l_1, l_2 \) is \( 2x^2 + 6xy + y^2 = 0 \) and that of the lines \( m_1, m_2 \) is \( 4x^2 + 18xy + y^2 = 0 \). If the angle between \( l_1 \) and \( m_2 \) be \( \alpha \)
(a) \( \frac{\pi}{2} - \alpha \)
(b) \( 2\alpha \)
(c) \( \frac{\pi}{4} + \alpha \)
(d) \( \alpha \)
Answer: (d) \( \alpha \)

Question. The lines represented by \( x^2 + 2\lambda xy + 2y^2 = 0 \) and the lines represented by \( (1 + \lambda)x^2 - 8xy + y^2 = 0 \) are equally inclined then \( \lambda \) is
(a) any real number
(b) greater than 2
(c) \( \pm 2 \)
(d) less than -2
Answer: (c) \( \pm 2 \)

Question. The area of the triangle formed by two rays whose combined equation is \( y = |x| \) and the line \( x + 2y = 2 \) is
(a) \( \frac{8}{3} \) unit\(^2 \)
(b) \( \frac{4}{3} \) unit\(^2 \)
(c) 4 unit\(^2 \)
(d) \( \frac{16}{3} \) unit\(^2 \)
Answer: (b) \( \frac{4}{3} \) unit\(^2 \)

Question. The centroid of the triangle whose three sides are given by the combined equation \( (x^2 + 7xy + 2y^2)(y - 1) = 0 \)
(a) \( \left( \frac{2}{3}, 0 \right) \)
(b) \( \left( \frac{7}{3}, \frac{2}{3} \right) \)
(c) \( \left( -\frac{7}{3}, \frac{2}{3} \right) \)
(d) none of the options
Answer: (c) \( \left( -\frac{7}{3}, \frac{2}{3} \right) \)

Question. The orthocenter of the triangle formed by the pair of lines \( 2x^2 - xy - y^2 + x + 2y - 1 = 0 \) and the line \( x + y + 1 = 0 \) is
(a) (-1, 0)
(b) (0, 1)
(c) (-1, 1)
(d) none of the options
Answer: (a) (-1, 0)

Question. The angle between the pair of lines whose equation is \( 4x^2 + 10xy + my^2 + 5x + 10y = 0 \) is
(a) \( \tan^{-1} \frac{3}{8} \)
(b) \( \tan^{-1} \frac{3}{4} \)
(c) \( \tan^{-1} \frac{2\sqrt{25 - 4m}}{m + 4}, m \in R \)
(d) none of the options
Answer: (b) \( \tan^{-1} \frac{3}{4} \)

Question. The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented by \( 2x^2 - xy - y^2 = 0 \) is
(a) \( 2x^2 - xy - 2y^2 + 4x - y = 6 \)
(b) \( 2x^2 - xy - y^2 - 4x - y + 2 = 0 \)
(c) \( 2x^2 - xy - y^2 - 4x + y + 2 = 0 \)
(d) none of the options
Answer: (c) \( 2x^2 - xy - y^2 - 4x + y + 2 = 0 \)

Question. The equation \( x^2 + (\lambda + \mu)xy + \lambda\mu y^2 + x + \mu y = 0 \) represents two parallel straight lines if
(a) \( \lambda + \mu = 0 \)
(b) \( \lambda = 4\mu \)
(c) \( \lambda = \mu \)
(d) none of the options
Answer: (c) \( \lambda = \mu \)

Question. The product of perpendiculars drawn from the point (1, 2) to the pair of lines \( x^2 + 4xy + y^2 = 0 \) is
(a) \( \frac{9}{4} \)
(b) \( \frac{3}{4} \)
(c) \( \frac{9}{16} \)
(d) none of the options
Answer: (a) \( \frac{9}{4} \)

Question. If pair of lines represented by \( ax^2 + 2hxy + by^2 = 0, b \ne 0 \), are such that the sum of the slopes of the lines is three the product of their slopes then
(a) \( 3b + 2h = 0 \)
(b) \( 3a + 2h = 0 \)
(c) \( 3h + 2a = 0 \)
(d) none of the options
Answer: (b) \( 3a + 2h = 0 \)

Question. The pair of lines \( \sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0 \) are rotated about the origin by \( \frac{\pi}{6} \) in the anticlockwise sense. The equation of the pair in the new position is
(a) \( \sqrt{3}x^2 - xy = 0 \)
(b) \( x^2 - \sqrt{3}xy = 0 \)
(c) \( xy - \sqrt{3}y^2 = 0 \)
(d) none of the options
Answer: (a) \( \sqrt{3}x^2 - xy = 0 \)

Question. The equation of the image of the pair of rays \( y = |x| \) by the line \( x = 1 \) is
(a) \( |y| = x + 2 \)
(b) \( |y| + 2 = x \)
(c) \( y = |x - 2| \)
(d) none of the options
Answer: (c) \( y = |x - 2| \)

Question. Two lines represented by the equation \( x^2 - y^2 - 2x + 1 = 0 \) are rotated about the point (1, 0), the line making the bigger angle with the positive direction of the x-axis being turned by 45° in the clockwise sense and the other line being turned by 15° in the anticlockwise sense. The combined equation of the pair of lines in their new positions is
(a) \( \sqrt{3}x^2 - xy + 2\sqrt{3}x - y + \sqrt{3} = 0 \)
(b) \( \sqrt{3}x^2 - xy - 2\sqrt{3}x + y + \sqrt{3} = 0 \)
(c) \( \sqrt{3}x^2 - xy - 2\sqrt{3}x + \sqrt{3} = 0 \)
(d) none of the options
Answer: (b) \( \sqrt{3}x^2 - xy - 2\sqrt{3}x + y + \sqrt{3} = 0 \)

Choose the correct options. One or more options may be correct.

Question. The equation \( x^3 + x^2y - xy^2 - y^3 = 0 \) represents
(a) three real straight lines
(b) lines in which two of them are perpendicular to each other
(c) lines in which two of them are coincident
(d) none of the options
Answer: (a) three real straight lines, (b) lines in which two of them are perpendicular to each other, (c) lines in which two of them are coincident

Question. The combined equation of two sides of an equilateral triangle is \( x^2 - 3y^2 - 2x + 1 = 0 \). If the length of a side of the triangle is 4 then the equation of the third side is
(a) \( x = 2\sqrt{3} + 1 \)
(b) \( y = 2\sqrt{3} + 1 \)
(c) \( x + 2\sqrt{3} = 1 \)
(d) \( x = 2\sqrt{3} \)
Answer: (a) \( x = 2\sqrt{3} + 1 \), (c) \( x + 2\sqrt{3} = 1 \)

Question. Two pairs of straight lines have the equations \( y^2 + xy - 12x^2 = 0 \) and \( ax^2 + 2hxy + by^2 = 0 \). One line will be common among them if
(a) \( a = -3(2h + 3b) \)
(b) \( a = 8(h - 2b) \)
(c) \( a = 2(b + h) \)
(d) \( a = -3(b + h) \)
Answer: (a) \( a = -3(2h + 3b) \), (b) \( a = 8(h - 2b) \)

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

Question. The straight lines represented by \( x^2 + mxy - 2y + 3y - 1 = 0 \) meet at
(a) \( \left( -\frac{1}{3}, \frac{2}{3} \right) \)
(b) \( \left( -\frac{1}{3}, -\frac{2}{3} \right) \)
(c) \( \left( \frac{1}{3}, \frac{2}{3} \right) \)
(d) none of the options
Answer: (a) \( \left( -\frac{1}{3}, \frac{2}{3} \right) \), (c) \( \left( \frac{1}{3}, \frac{2}{3} \right) \)

Question. If the chord \( y = mx + 1 \) of the circle \( x^2 + y^2 = 1 \) subtends an angle of measure 45° at the major segment of the circle then the value of m is
(a) 2
(b) 1
(c) -1
(d) none of the options
Answer: (b) 1, (c) -1

Question. The equation \( 2x^2 - 3xy - py^2 + x + qy - 1 = 0 \) represents two mutually perpendicular lines if
(a) \( p = 3, q = 2 \)
(b) \( p = 2, q = 3 \)
(c) \( p = -2, q = 3 \)
(d) \( p = 2, q = -\frac{9}{2} \)
Answer: (b) \( p = 2, q = 3 \), (d) \( p = 2, q = -\frac{9}{2} \)

Question. The diagonals of a square are along the pair of lines whose equation is \( 2x^2 - 3xy - 2y^2 = 0 \). If (2, 1) is a vertex of the square then another vertex consecutive to it can be
(a) (1, -2)
(b) (1, 4)
(c) (-1, 2)
(d) (-1, -4)
Answer: (a) (1, -2), (c) (-1, 2)

Question. There is a pair of points, one on each of the lines whose combined equation is \( (4x - 3y + 5)(6x + 8y + 5) = 0 \). If they are such that the distance of the point on one line is 2 units from the other line then the points are
(a) \( \left( -\frac{1}{10}, \frac{9}{5} \right) \) and \( \left( \frac{1}{2}, -1 \right) \)
(b) \( \left( \frac{1}{2}, -1 \right) \) and \( \left( -\frac{23}{10}, -\frac{7}{3} \right) \)
(c) \( \left( -\frac{1}{10}, \frac{9}{5} \right) \) and \( \left( -\frac{23}{10}, -\frac{7}{5} \right) \)
(d) none of the options
Answer: (a) \( \left( -\frac{1}{10}, \frac{9}{5} \right) \) and \( \left( \frac{1}{2}, -1 \right) \), (b) \( \left( \frac{1}{2}, -1 \right) \) and \( \left( -\frac{23}{10}, -\frac{7}{3} \right) \)

Question. The pairs of straight lines \( ax^2 + 2hxy - ay^2 = 0 \) and \( hx^2 - 2axy - hy^2 = 0 \) are such that
(a) one pair bisects the angles between the other pair
(b) the lines of one pair are equally inclined to the lines of the other pair
(c) the lines of one pair are perpendicular to the lines of the other pair
(d) none of the options
Answer: (a) one pair bisects the angles between the other pair, (b) the lines of one pair are equally inclined to the lines of the other pair

Question. If the pair of lines \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) intersect on the y-axis then
(a) \( 2fgh = bg^2 + ch^2 \)
(b) \( bg^2 = ch^2 \)
(c) \( abc = 2fgh \)
(d) none of the options
Answer: (a) \( 2fgh = bg^2 + ch^2 \), (b) \( bg^2 = ch^2 \)

Question. The combined equation of three sides of a triangle is \( (x^2 - y^2)(2x + 3y - 6) = 0 \). If (-2, a) is an interior point and (b, 1) is an exterior point of the triangle then
(a) \( 2 < a < \frac{10}{3} \)
(b) \( -2 < a < \frac{10}{3} \)
(c) \( -1 < b < \frac{9}{3} \)
(d) \( -1 < b < 1 \)
Answer: (a) \( 2 < a < \frac{10}{3} \), (d) \( -1 < b < 1 \)