Pair of Straight Lines Questions with Solutions

9 questions2 easy4 medium3 hard

A pair of straight lines can be represented by a single second-degree equation. This topic covers the homogeneous equation ax^2 + 2hxy + by^2 = 0, which represents two lines through the origin, and the general equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Exam questions test the condition for the general equation to represent a pair of lines, the angle between the lines (tan theta = 2 sqrt(h^2 - ab) / |a + b|), the conditions for the lines to be perpendicular (a + b = 0) or coincident (h^2 = ab), and the sum and product of the slopes (-2h/b and a/b). Knowing these results directly turns most questions into short calculations. Practise the pair of straight lines questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
If two lines represented by $x^{4} + x^{3} y + c x^{2} y^{2} - x y^{3} + y^{4} = 0$ bisect the angle between the other two, then the value of c is
A. 0B. -1C. 1D. -6
hardAsked in MAH MCA CET 2024View solution ›
Q2.
If the two pairs of lines $x^{2} - 2 m x y - y^{2} = 0$ and $x^{2} - 2 n x y - y^{2} = 0$ are such that one of them represents the bisectors of the angles between the other, then
A. $mn + 1 = 0$ B. $mn - 1 = 0$ C. $\frac{1}{m} + \frac{1}{n} = 0$ D. $\frac{1}{m} - \frac{1}{n} = 0$
hardAsked in NIMCET 2010View solution ›
Q3.
If $y = m x$ bisects the angle between the lines $x^{2} (t an^{2} θ + cos^{2} θ) + 2 x y t an θ - y^{2} s in^{2} θ = 0$ when $θ = \frac{π}{3}$ , then the value of $3 m^{2} + 4 m$ is
A. $1$ B. $\frac{1}{3}$ C. $3$ D. $73$
hardAsked in NIMCET 2008View solution ›
Q4.
The acute angle between the pair of lines $x^{2} - 3 x y + 2 y^{2} = 0$ is
A. $t an^{- 1} (\frac{1}{3})$ B. $t an^{- 1} (\frac{1}{2})$ C. $t an^{- 1} (3)$ D. $\frac{π}{4}$
mediumView solution ›
Q5.
The lines given by $k x^{2} + 4 x y - 5 y^{2} = 0$ are perpendicular to each other if $k$ equals
A. −5B. 4C. 5D. −4
easyView solution ›
Q6.
If $θ$ is the acute angle between the pair of lines $x^{2} - 7 x y + 12 y^{2} = 0$ , then $\frac{2 cos θ + 3 s in θ}{4 s in θ + 5 cos θ} =$
A. $\frac{29}{69}$ B. $\frac{61}{39}$ C. $\frac{39}{61}$ D. $\frac{69}{29}$
mediumAsked in NIMCET 2021View solution ›
Q7.
If $x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0$ represents a pair of straight lines, their point of intersection is
A. (0, 0)B. (8, 5)C. (8, −5)D. (−2, 5)
mediumAsked in NIMCET 2017View solution ›
Q8.
The slopes of the two lines $6 x^{2} - x y - 2 y^{2} = 0$ differ by
A. $\frac{5}{2}$ B. $\frac{7}{2}$ C. $\frac{7}{5}$ D. $- 1$
easyAsked in NIMCET 2018View solution ›
Q9.
For what values of $λ$ does the equation $6 x^{2} - x y + λ y^{2} = 0$ represent two perpendicular lines and two lines inclined at an angle of $\frac{π}{4}$ , respectively?
A. -6 and -2B. 6 and 1C. -6 and -35D. -6 and 1
mediumAsked in NIMCET 2024View solution ›